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Trigonometrical Ratios, Functions and Identities, , 31, , Measurement of Angles, Trigonometrical Ratios , Function and, Identities, , Basic Level, , 1., , Which of the following relation is correct, (c) sin 1  sin 1o, , (b) sin 1  sin 1o, , (a) sin 1o  sin 1, , [WB JEE 1991], , (d), , , 180, , sin 1  sin 1o, , 2., , The radius of the circle whose arc of length 15 cm makes an angle of 3/4 radian at the centre is, 1, 1, (a) 10 cm, (b) 20 cm, (c) 11 cm, (d) 22 cm, 4, 2, , 3., , If tan  , , 4, , then sin  , 3, , [Orissa JEE 2002; IIT, , 1979], , (a), 4., , 4, 4, but not, 5, 5, , (b) , , (c), , 4, 4, but not , 5, 5, , (d) None of these, , If f (x )  cos 2 x  sec 2 x , then, (b) f ( x )  1, , (a) f (x )  1, 5., , 4, 4, or, 5, 5, , If x  sec   tan  , then x , , (d) f ( x )  2, , (c) 2, , (d) 2 tan , , 1, , x, , (b) 2 sec , , (a) 1, , (c) 1  f ( x )  2, , 6., , If A lies in the second quadrant and 3 tan A  4  0 then the value of 2 cot A  5 cos A  sin A is equal to [Harayana CEE 199, 7, 23, 7, 53, (a), (b), (c), (d), 10, 10, 10, 10, , 7., , tan 1o tan 2o tan 3o tan 4 o.......... .. tan 89 o , (a) 1, (b) 0, The incorrect statement is, 1, (a) sin   , (b) cos   1, 5, , 8., , 9., , 2 cos , , 2 sin , , (b), , 1, 2, , (d) tan   20, [WB JEE 1988], , (c) 2 cos , , (d)  2 cos , [MP PET 1994], , p 1, 2p, , 2p, , (b), , p 1, 2, , p 1, 2p, 2, , 2, , (c), , If sin   cos   1, then sin  cos  , (a) 0, , 12., , (c) sec  , , If sec   tan   p, then tan  is equal to, (a), , 11., , (d) 1/2, , If cos   sin   2 sin  , then cos   sin  is equal to, (a), , 10., , [MP PET 1998, 2001], , (c) , , o, , o, , The value of cos 1 cos 2 cos 3 ..... cos 179, , 2p, p 1, 2, , [Karnataka CET 1998], , (b) 1, o, , (d), , (c) 2, o, , is, , (d) 1/2, [Karnataka CET 1999]

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32 Trigonometrical Ratios, Functions and, Identities, (a), 13., , (b) 0, , 2, , If tan   , , 1, 5, , (b), , 19., , 20., , 21., , 22., , 24, 5, , (c), , 24, 5, , 5, , (d) , , 6, [EAMCET 1994], , (d), , 48, 5, [Karnataka CET 1998], , (c) sec  . cosec , , (b) 1, , If tan  , , (d) sin 2   cos 2 , , 20, , cos  will be, 21, , 20, 41, , If cosec A  cot A , (a), , 18., , 5, , (c), , (sec 2   1) (cosec 2  1) =, , (a) , 17., , (d) None of these, , 6, 6, 6, If A lies in the third quadrant and 3 tan A  4  0, then 5 sin 2 A  3 sin A  4 cos A , , (a) 0, 16., , 1, , (b) , , (a) 0, 15., , (c) 1, , and  lies in the 1st quadrant, then cos  is, , 1, , (a), 14., , 1, , 21, 22, , [MP PET 1994], , (b) , , 1, 21, , (c) , , 21, 29, , (d) , , 20, 21, , 11, , then tan A equal to, 2, , (b), , [Roorkee 1995], , 15, 16, , (c), , 44, 117, , 24, and  lies in the second quadrant, then sec   tan  equal to, 25, (a) – 3, (b) – 5, (c) –7, 5 sin   3 cos , If 5 tan   4 , then, equal to, 5 sin   2 cos , (a) 0, (b) 1, (c) 1/6, 1  cos , equal to, sin 2 , 1, (a) 0, (b) 1, (c), 1  cos , , (d), , 117, 43, , If sin  , , 1, simplifies to, tan A  cot A, (a) sec A cosec A, (b) sin A cos A, , [MP PET 1997], , (d) – 9, [Karnataka CET 1998], , (d) 6, [Karnataka CET 1998], , (d), , 1, 1  cos , , The expression, , [SCRA 1999], , (c) tan 2 A, , 1, If for real values of x, cos   x  , then, x, (a)  is an acute angle (b)  is a right angle, , (d) sin 2 A, [MP PET 1996], , (c)  is an obtuse angle, , (d) No, , value, , of, , , , is, , possible, 23., , If sin x + cosec x =2, then sinn x  cosec n x is equal to, (a) 2, , (b) 2, , n, , [UPSEAT 2002], , (c) 2, , n 1, , (d) 2, , n2, , Advance Level, 24., , One root of the equation cos x  x ,  , (a) 0, ,  2, , 1,  0 lies in the interval, 2, ,   , (b)   , 0 ,  2 , ,  , (c)  ,  , 2 , ,  3 , (d)  ,, 2 , 

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Trigonometrical Ratios, Functions and Identities, 25., , If, , (a), 26., , 1, y, , (c) 1 – y, , (d) 1 + y, , If sin   sin 2   sin 3   1 , then cos 6   4 cos 4   8 cos 2  =, (b) 2, , (c) 1, , (d) None of these, , If  and  are angles in the 1 quadrant such that tan   1 / 7 and sin   1 / 10 . Then, st, , (a)   2  90 o, , 28., , [BIT Ranchi 1996], , (b) y, , (a) 4, 27., , 33, , {1  cos   sin  }, 2 sin , = y, then, =, 1  sin , {1  cos   sin  }, , (b)   2  60 o, , (c)   2  30 o, , (d)   2  45 o, , 1  sin 2 , cos 2 , 4 sin 4, 2, The value of  lying between 0 and  / 2 and satisfying the equation sin , 1  cos 2 , 4 sin 4  0, 2, 2, sin , cos , 1  4 sin 4, [IIT 1988; MNR 1992; Kurukshetra CEE 1998; DCE 1996], , 11, 7, (a), or, 24, 24, , 29., , If, , 3,     , then, 4, , 5, (b), 24, , If for all real values of x ,, , If tan  , , (a), 32., , [Pb. CET 2000, AMU 2001], , (c) 1  cot , , (b) 1  cot , ,  , (a)  0, ,  3, , 31., , (d) None of these, , 24, , co sec 2  2 cot  is equal to, , (a) 1  cot , 30., , (c), , , , 4x2  1, 64 x  96 x . sin   5, 2, , , , (d) 1  cot , , 1, , then  lies in the interval, 32, ,   2 , (b)  ,, , 3 3 , ,  2, , , , (c) , 3, , , , [Roorkee 1998], ,  4  5 , ,, (d) , ,  3 3 , , 3, , then the sum of the infinite series 1  2(1  cos  )  3(1  cos  )2  4 (1  cos  )3  ....  is, 2, , 2, 3, , (b), , 3, 4, , (c), , 5, 2 2, , (d), , 5, 2, , Let A0 A1 A2 A3 A4 A5 be a regular hexagon inscribed in a circle of unit radius. Then the product of the lengths of, the line segments A0 A1 , A0 A 2 and A 0 A 4 is, (a), , 3, 4, , (b) 3 3, , (c) 3, , (d), , 3 3, 2, , Trigonometrical Ratios of Allied Angles, , Basic Level, 33., , If x sin 45 o cos 2 60 o , , tan 2 60 o cosec 30 o, sec 45 o cot 2 30 o, , (a) 2, 34., , (b) 4, , (c) 8, , (d) 16, , (b) 0, , [MP PET 1990], , (c) 1, , (d) None of these, , sin(   ) sin(   )cosec  , 2, , (a) 1, 36., , [Kerala (Engg.) 2002], , cos A  sin(270 o  A)  sin(270 o  A)  cos(180 o  A) , , (a) –1, 35., , , then x , , [EAMCET 1980], , (b) –1, , The value of sin 600 o cos 330 o  cos 120 o sin 150 o is, , (c) sin , , (d)  sin , [MP PET 1994]

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34 Trigonometrical Ratios, Functions and, Identities, (a) –1, , (b) 1, , 1, , (c), , 3, 2, , (d), , 2, , 37., , If A  130, , o, , and x  sin A  cos A, then, (b) x  0, , (a) x  0, 38., , [Karnataka CET 1989], , (d) x  0, , (c) x  0, , , , , , tan  sin    cos     , 2, 2, , , , , , (a) 1, , [EAMCET 1981], , (b) 0, , 1, , (c), , (d) None of these, , 2, , 39., , sin 2 5 o  sin 2 10 o  sin 2 15 o  ......  sin 2 85 o  sin 2 90 o , , [Karnataka, , CET, , 1999,, , 1995], , (a) 7, 40., , (b) 8, , The value of, , cot 54, tan 36, , o, , is, , cot 70 o, , [Karnataka CET 1999], , (b) 3, , (c) 1, , (d) 0, , (c) 4, , (d) 8, , (c) –1, , (d) 2, , [Roorkee 1989], , (b) 2, , 25, , 24, , [Karnataka CET 2003], , (b) 1, , If tan( A  B)  1, sec( A  B) , , 2, 3, , (b), , , then the smallest positive value of B is, 19, , 24, , (c), , [Kerala (Engg.) 2002], , 13, , 24, , (d), , 11, , 24, , If x  sin 130 o cos 80 o , y  sin 80 o cos 130 o , z  1  xy , which one of the following is true, (a) x  0, y  0, z  0, , 47., , tan 20, , (d) – 4, , cos 1o  cos 2o  cos 3o  ........  cos 180 o , , (a), 46., , , , (c) – 3, , o, , tan 9 o  tan 27 o  tan 63 o  tan 81 o , , (a) 0, 45., , [MP PET 1997], , (b) – 2, o, , (a) 1/2, 44., , (d) 210 o ,330 o, , The value of tan( 945 o ) is, , (a) 2, 43., , 1, 2, [EAMCET 1994], , (c) 210 o , 240 o, , (b) 240 o , 300 o, , (a) –1, 42., , (d) 9, , Values of  (0    360 o ) satisfying cosec   2  0 are, (a) 210 o , 300 o, , 41., , (c) 9, , (b) x  0, y  0,0  z  1, , [AMU 1999], , (d) x  0, y  0, 0  z  1, , (c) x  0, y  0, z  1, , If   22 o 3 0 , then (1  cos ) (1  cos 3)(1  cos 5)(1  cos 7) equals, (a) 1/8, , (b) 1/4, , (c), , [AMU 1999], , 1 2, , (d), , 2 2, , 2 1, 2 1, , Trigonometrical Ratios of Sum & Difference of Two Angles, Tranformation of Product into Sum &, Difference, Transformation of Sum & Difference into Product, , Basic Level, 48., , If A, B, C, D are the angles of a cyclic quadrilateral then cos A  cos B  cos C  cos D , (a) 2(cos A  cos C), , 49., , cos 17 o  sin 17 o, cos 17 o  sin 17 o, 1982)], , (b) 2(cos A  cos B), , (c) 2(cos A  cos D), , [IIT 1970], , (d) 0, [MP PET 1998 (Similar to EAMCET

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Trigonometrical Ratios, Functions and Identities, , 50., , cot(45 o   ) cot(45 o   ) , , (a) –1, 51., , tan 75  cot 75, , o, , [MNR 1982], , 3 cosec 20  sec 20 , (b), , 3, 2, , 1, 2, , (c) (1,  1), , (d) (1, 1), [MP PET 1993], , (c), , 1, 2, , (d) 0, [MNR 1975], , (c), , (d), , 2, , 1, , cos 2 48 o  sin 2 12 o , 5 1, 4, , [MNR 1977], , 5 1, 8, , (b), , 3 1, 4, , (c), , 3 1, , (d), , 2 2, , sin 20 o sin 40 o sin 60 o sin 80 o , , [MNR 1976, 1981], , (b) 5 / 16, , (d) 5/16, , (c) 3/16, , cos 20 o cos 40 o cos 80 o , , cos, , [MP PET 1989], , (b) 1/4, , (c) 1/6, , (d) 1/8, , 2, 4, 8, 16 , cos, cos, cos, , 15, 15, 15, 15, , [IIT 1985], , (b) 1/4, , (c) 1/8, , (d) 1/16, , If x  cos 10 cos 20 cos 40 , then the value of x is, o, , o, , 1, tan 10 o, 4, , o, , (b), , [Roorkee 1995], , 1, cot 10 o, 8, , (c), , 1, cosec 10 o, 8, , (d), , 1, sec 10 o, 8, , The value of cos 52 o  cos 68 o  cos 172 o is, , [MP PET 1997], , (b) 1, , 3, 2, , (c) 2, , (d), , 1, (c), 2, , 1, (d) , 2, , cos 40 o  cos 80 o  cos 160 o  cos 240 o , , [EAMCET 1996], , (b) 1, , 1  cos 56 o  cos 58 o  cos 66 o , o, , o, , o, , [IIT 1964], o, , o, , (a) 2 cos 28 cos 29 cos 33 (b) 4 cos 28 cos 29 cos 33, 65., , (d) sin 15  cos 15 o, o, , o, , 2, , (a) 0, 64., , sin 40 o, , [MP PET 1992], , (b) 1, , (a) 0, 63., , 4 sin 20 o, , The value of cos 105 o  sin 105 o is, , (a), 62., , (c) cos 15  sin 15, , (b) 1, , (a) 1/2, 61., , (d), , o, , (b) (1, 1), , (a) 1/2, 60., , (b) 2 sin 15, , o, , (c) 4, , [MP PET 1992], , (a) 3/16, 59., , sin 40, , o, , cos 2   cos 2 (  120 o )  cos 2 (  120 o ) is equal to, , (a), 58., , 2 sin 20 o, , If cos( A  B)   cos A cos B   sin A sin B, then ( ,  ) , , (a), 57., , [IIT 1988], , sin 15 o  cos 105 o , , (a), 56., , (d) None of these, , o, , (a) (1,  1), 55., , (c) 2  3, , (b) 2  3, o, , (a) 0, 54., , (d) , , (c) 1, , , , (a) 2, 53., , [MNR 1973], , (b) 0, , o, , (a) 2 3, 52., , 35, , (d) tan 73 o, , (c) tan 54 o, , (b) tan 56 o, , (a) tan 62 o, , cos 15 o , 1978], , o, , o, , o, , (c) 4 cos 28 cos 29 sin 33, , o, , (d) 2 cos 28 cos 29 sin 33 o, [MP, , o, , o, , PET, , 1998;, , MNR

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36 Trigonometrical Ratios, Functions and, Identities, 1  cos 30 o, 2, , (a), 66., , 1  cos 30 o, 2, , (b), , (c) , , 1  cos 30 o, 2, [EAMCET 1991], , (c) 0, , (d) None of these, , If cos   cos   0  sin   sin  , then cos 2  cos 2  , (a) 2 sin(   ), , 68., , (d) , , tan 5 x tan 3 x tan 2 x , sin 5 x  sin 3 x  sin 2 x, (a) tan 5 x  tan 3 x  tan 2 x (b), cos 5 x  cos 3 x  cos 2 x, , 67., , 1  cos 30 o, 2, , If tan A  , , (b) 2 cos(   ), , [EAMCET 1994], , (d) 2 cos(   ), , (c) 2 sin(   ), , 1, 1, and tan B   , then A+B =, 3, 2, , [IIT 1967; UPSEAT 1987; MP PET, , 1989], , (a), 69., , , 4, , If cos( A  B) , , 1, 5, , 5, 4, , (d) None of these, , (b) sin A sin B  , , [MP PET 1997], , 2, 5, , (c) cos A cos B  , , 1, 5, , (d) sin A sin B  , , 1, 5, , sin 3  cos 3, 1 =, sin   cos , , (b) 2 cos 2, , (a) 2 sin 2, 71., , (c), , 3, and tan A tan B  2, then, 5, , (a) cos A cos B , 70., , 3, 4, , (b), , (c) tan 2, , (d) cot 2, , tan 3 A  tan 2 A  tan A =, , [MNR 1982;, , Pb., , CET, , 1991], , 72., , 73., , (a) tan 3 A tan 2 A tan A, , (b)  tan 3 A tan 2 A tan A, , (c) tan A tan 2 A  tan 2 A tan 3 A  tan 3 A tan A, , (d) None of these, , If cos A  m cos B, then, (a) cot, , A  B m 1, BA, , tan, 2, m 1, 2, , (b) tan, , (c) cot, , A  B m 1, AB, , tan, 2, m 1, 2, , (d) None of these, , (b) 1, , [Kerala CEE 1993], , (c) –1/2, , (d) 1/8, , tan 100 o  tan 125 o  tan 100 o tan 125 o , (a) 0, , 75., , A  B m 1, BA, , cot, 2, m 1, 2, , The value of cos 12 o  cos 84 o  cos 156 o  cos 132 o is, (a) 1/2, , 74., , [UPSEAT 1990], , 1, 2, , (b), , If cos P , , [DCE 1999], , (c) –1, , (d) 1, , 1, 13, and cos Q , where P and Q both are acute angles. Then the value of P – Q is, 7, 14, [Orissa JEE 2002; Karnataka CET 2002], , (a) 30, 76., , o, , If sin A , , (b) 60, , 1, , and sin B , , 1, , (c) 45, , o, , (d) 75 o, , , where A and B are positive acute angles, then A  B , , 5, , 10, (a) , , o, , (b), , , 2, , (c), , , 3, , (d), , , 4

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Trigonometrical Ratios, Functions and Identities, 77., , 37, ,     3 , sin  sin, ,  10   10 , , [MNR 1984], , (a) 1/2, 78., , (b) –1/2, , AB, 0, 2, , (c) cos, , AB, 0, 2, , (d) cos( A  B)  0, , (c) 1/8, , (d) 1/4, , If (1  tan  )(1  tan  )  2 , then    =, (b) 45o, , [Karnataka CET 1993], , (c) 60o, , (d) 75o, , , , , , cos 2      sin 2     , 6, 6, , , , , 1, cos 2, 2, , [EAMCET 2001], , (c) , , (b) 0, , 1, cos 2, 2, , (d), , 1, 2, , If sin   sin 2  sin 3  sin  and cos   cos 2  cos 3  cos  , then  is equal to, (b) , , (c) 2, , [AMU 2001], , (d)  / 6, , cos  . sin(   )  cos  . sin(   )  cos  . sin(   ) , , (a) 0, 85., , [EAMCET 1994], , [IIT 1982], , (b) 1/32, , (a)  / 2, 84., , (d) 2, , sin 12 o sin 48 o sin 54 o , , (a), 83., , AB, 0, 2, , (b) sin, , (a) 30o, 82., , (c) 1/2, , If sin A  sin B and cos A  cos B , then, , (a) 1/16, 81., , [MNR 1979], , (b) 0, , (a) sin, 80., , (d) 1, , sin 50 o  sin 70 o  sin 10 o , , (a) 1, 79., , (c) 1/4, , (b) 1/2, , [EAMCET 2003], , (c) 1, , (d) 4 cos  cos  cos , , , ,    ,    , Given that cos , is equal to,   2 cos ,  , then tan tan, 2, 2,  2 ,  2 , 1, 2, , (a), , (b), , 1, 3, , (c), , 1, 4, , (d), , 1, 8, , Advance Level, 86., , If sin A  sin B  C, cos A  cos B  D, then the value of sin( A  B) , (a) CD, , 87., , (b), , If A  B  225 o , then, (a) 1, , 88., , o, , –, , (c), , C 2  D2, 2 CD, , (d), , 2CD, C 2  D2, , cot A, cot B, ., , 1  cot A 1  cot B, , [MNR 1974], , (c) 0, , (d), , 1, 2, , 3, , [IIT 1974], , cos 10 o, , (a) 0, 89., , C 2  D2, , (b) –1, , 1, sin 10, , CD, , (b) 1, , (c) 2, , (d) 4, , sin 3  sin 5  sin 7  sin 9, , cos 3  cos 5  cos 7  cos 9, , (a) tan 3, , (b) cot 3, , [Roorkee 1973], , (c) tan 6, , (d) cot 6

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38 Trigonometrical Ratios, Functions and, Identities, 90., , If cos(  ), cos  and cos(   ) are in H.P., then cos  sec, , 91., , cos B  sin B, cos B  sin B, , (b), , n 1, n 1, , (b), , If 3 sin   5 sin  , then, , tan, tan, , (a) 1, 94., , If, , , 2, , (a), 95., , (d) None of these, , (c), , cos A  sin A, cos A  sin A, , (d) None of these, , (c), , 1n, n 1, , (d), , tan( x  y), is, tan( x  y), , n 1, n 1, , 1n, 1n, , , 2, , , , =, , [EAMCET 1996], , 2, , (c) 3, , (d) 4, , 12, 15, 3, , the value of sin(    ) is, , sin  , and tan  , 17, 5, 2, , 171, 221, , (b), , 21, 221, , (c), , 21, 221, , [Roorkee 2000], , (d), , 17, 221, , cos 2 76 o  cos 2 16 o  cos 76 o cos 16 o , 1, 4, , The value of cos , , (a), 97., , cos A  sin A, cos A  sin A, , (b) 2, ,    ,    , , (a) , 96., , [IIT 1997], , [Roorkee 1970; IIT 1966], , If sin 2 x  n sin 2 y, then the value of, (a), , 93., , is equal to, , sin(B  A)  cos( B  A), =, sin(B  A)  cos( B  A), , (a), 92., , 2, , (c)  1 / 2, , (b)  3, , (a)  2, , , , (b), , , 12, , 3, 2, ,  cos 2, , , 4, , 1, 2, ,  cos 2, , (b), , [EAMCET 2002], , (c) 0, , (d), , 3, 4, , 5, is, 12, , 2, 3, , [Karnataka CET 2002], , (c), , 3 3, 2, , (d), , 2, 3 3, , If angle  be divided into two parts such that the tangents of one part is K times the tangent of the other and, ,  is their difference, then sin  =, (a), 98., , 99., , K 1, sin , K 1, , (b), , K 1, sin , K 1, , (c), , 2K  1, sin , 2K  1, , (d) None of these, , If tan  , tan  are the roots of the equation x 2  px  q  0 (p  0) ,then, p, q 1, , (a) sin 2 (   )  p sin(   ) cos(   )  q cos 2 (   )  q, , (b) tan (   ) , , (c) cos(   )  1  q, , (d) sin(   )   p, , tan  equals the integral solution of the inequality 4 x 2  16 x  15  0 and cos  equals to the slope of the, , If, , bisector of first quadrant, then sin(   ) sin(   ) is equal to, (a), , 100., , 3, 5, , 2  sin   cos , , sin   cos , , (b), , 3, 5, , (c), , 2, 5, , (d), , 4, 5, [AMU 1999]

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Trigonometrical Ratios, Functions and Identities,   , (a) sec   , 2 8, ,   , (b) cos   , 8 2, ,   , (c) tan   , 2 8, , 101. The sum S  sin   sin 2  .......  sin n , equals, (a) sin, , 1, 1, , (n  1) sin n  / sin, 2, 2, 2, , (b) cos, , 1, 1, , (n  1) sin n  / sin, 2, 2, 2, , (c) sin, , 1, 1, , (n  1) cos n  / sin, 2, 2, 2, , (d) cos, , 1, 1, , (n  1) cos n  / sin, 2, 2, 2, , 39, ,   , (d) cot   , 2 2, , [AMU 2002], , Trigonometrical Ratios of Multiple and Sub-multiple of an Angle, , Basic Level, , 102., , 2 cos 2   2 sin 2   1, then  , , (b) 30 o, , (a) 15o, , (c) 45 o, , (d) 60 o, , A, 5, 3, , then 32 sin cos A , 2, 2, 4, , 103. If cos A , (a), , [Karnataka CET 1998], , (b)  7, , 7, , [EAMCET 1982], , (c) 7, , (d) –7, , 104. cot x  tan x , (b) 2 cot 2 x, , (a) cot 2 x, 105., , (c) 2 cot 2 x, , (d) cot 2 2 x, , (c) 1, , (d) None of these, , cos 2 A(3  4 cos 2 A)2  sin 2 A(3  4 sin 2 A)2 , , (a) cos 4 A, 106., , [MP PET 1986], , (b) sin 4 A, , 2 sin   4 cos(   ) sin  sin   cos 2(   ) =, 2, , (a) sin 2, 107. If tan A , , (b) cos 2 , , [UPSEAT 1993], , (c) cos 2, , (d) sin 2 , , 1  cos B, , then the value of tan 2 A in terms of tan B, sin B, , (a) tan 2 A  tan B, , (b) tan 2 A  tan 2 B, , 108. If tan A , , 1, 1, , tan B  , then cos 2A =, 3, 2, , (a) sin B, , (b) sin 2B, , (c) tan 2 A  tan 2 B  2 tan B, , (d) None of these, [Karnataka CET 1986, 89], , (c) sin 3B, , (d) None of these, , 109. If a cos 2  b sin 2  c has  and  as its solution, then the value of tan   tan  is, (a), 110., , ca, 2b, , (b), , (c), , ca, 2b, , (d), , b, ca, , 3 cos   cos 3, is equal to, 3 sin   sin 3, , (a) 1  cot 2 , 111., , 2b, ca, , [Haryana CEE 1998], , sin 2, , , 8, ,  sin 2, , [EAMCET 1996], , (b) cot 4 , , 3, 5, 7,  sin 2,  sin 2, =, 8, 8, 8, , (c) cot 3 , , (d) 2 cot , [Karnataka CET 1998]

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40 Trigonometrical Ratios, Functions and, Identities, (a) 1, , (b) –1, , 112. If k  sin, , , 18, , . sin, , cos, , , , cos, , 7, , (b) 1/8, , (a), 115., , [IIT 1993; UPSEAT 1974], , (c) 1/16, , (d) None of these, , 2, 4, =, cos, 7, 7, , (a) 0, , [MP PET 1998], , (b), , 114. If cos  , , (d) 2, , 5, 7, , then the numerical value of k is, . sin, 18, 18, , (a) 1/4, 113., , (c) 0, , 1, 2, , (c), , 1, 4, , (d) , , 1, 8, , 1, 1,  a   , then the value of cos 3 is, 2, a, , 1 3, 1, a , 8 , a3, , , , , , (b), , 3, 1, a  , 2, a, , [MP PET 2001], , (c), , 1 3, 1, a , 2 , a3, , , , , , 2 sin A cos 3 A  2 sin3 A. cos A , , (d), , 1 3, 1, a , 3 , a3, , , , , , [Roorkee 1975, Kerala (Engg.), , 2002], , (a) sin 4 A, 116. If cos A , , (b), , (c), , 1, sin 4 A, 4, , (d) None of these, , 3,  A   5A , ,then 32 sin  sin, , 4, 2  2 , , (a) 7, 117., , 1, sin 4 A, 2, , (b) 8, , [DCE 1996], , (c) 11, , (d) None of these, , If  is a root of 25 cos 2   5 cos   12  0, / 2     , then sin 2 is equal to, (a) 24/25, , (b) –24/25, , (c) 13/18, , [UPSEAT 2001], , (d) –13/18, , Advance Level, , 118., , tan 20 o tan 40 o tan 60 o tan 80 o , (a) 1, , (b) 2, , 119. If cos  , , (c) 3, , (d), , 3 /2, ,  , 4, 3, , and cos   , where  and  are positive acute angles, then cos, 5, 5, 2, , 7, , (a), , [IIT 1974], , (c), , 7, , 5 2, , 2, , 120. If cos(   ) , , 7, , (b), , 5, , 4, 5, , , sin(   ) , and ,  lie between 0 and , then tan 2 , 4, 5, 13, , [MP PET 1988], , (d), , 7, 2 5, [IIT, , 1979;, , EAMCET, , 2002], , (a), , 16, 63, , (b), , 56, 33, , (c), , 28, 33, , (d) None of these, , 2 , 4 , 1 1 1, , , 121. If x cos   y cos  ,   is equal to,   z cos  , , then the value of, 3, 3, x y z, , , , , , (a) 1, , (b) 2, , (c) 0, , 122. If a tan   b, then a cos 2  b sin 2 =, , [IIT 1984], , (d) 3 cos , [EAMCET 1981, 82; MP PET, , 1996], , (a) a, , (b) b, , (c) – a, , (d) – b

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Trigonometrical Ratios, Functions and Identities, , 41, , 123. If 2 sec 2  tan   cot  , then one of the values of  +  is, (a), , , 4, , (b), , , 2, , (c) , , (d) 2, , xy, 124. If cos x  cos y  cos   0 and sin x  sin y  sin   0, then cot , ,  2 , , (a) sin , 125. If sin 2  sin 2 , (a), , xy, (d) sin, ,  2 , , (c) cot , , (b) cos , , [Karnataka CET 2001], , 3, 1, and cos 2  cos 2  , then cos 2 (   ) , 2, 2, , 3, 8, , (b), , 5, 8, , (c), , [MP PET 2000], , 3, 4, , (d), , 5, 4, , 126. If (sec   tan )(sec   tan  )(sec   tan  )  tan  tan  tan  , then (sec   tan  )(sec   tan  )(sec   tan  ) =, [Haryana CEE 1998], , (a) cot  cot  cot , 127. If cos 2 B , , (b) tan  tan  tan , , (c) cot   cot   cot , , (d) tan   tan   tan , , cos( A  C), , then tan A, tan B, tan C are in, cos( A  C), , (a) A.P, , (b) G.P, , (c) H.P, , (d) None of these, ,  , 3  , 5  , 7 , , 128.  1  cos   1  cos,   1  cos,   1  cos, , 8, 8, 8 , 8,  ,  , ,  , (a), , 129. If, , (b), , 1, 4, , (c), , 1, 8, , (d), , 1, 16, , sin 4 A cos 4 A, 1, sin 8 A cos 8 A, , , , then the value of, is equal to, , a, b, ab, a3, b3, , (a), 130., , 1, 2, , [IIT 1994; WB JEE 1992], , 1, , (b), , (a  b)3, , a3b 3, (a  b)3, , (c), , a 2b 2, (a  b)2, , [WB JEE 1971], , (d) None of these, , 2  3  4  6 is equal to, , (a) cot 7, , 1, 2, , o, , (b) sin 7, , [IIT 1966, 1975], , 1, 2, , o, , (c) sin 15 o, , (d) cos 15 o, , 131. If sin  is the geometric mean between sin  and cos  , then cos 2  is equal to, , , (b) 2 cos 2    , 4, , , , , , (a) 2 sin 2    , 4, , , , , , (c) 2 cos 2    , 4, , , , , , (d) 2 sin 2    , 4, , , , 132. The value of k , for which (cos x  sin x )2  k sin x cos x  1  0 is an identity, is, (a) –1, , (b) – 2, , (c) 0, , [Kerala (Engg.) 2001], , (d) 1, , n, , 133. If sin 3 x sin 3 x , , c, , m, , cos mx where c0 , c1, c2 ,......, cn are constants and cn  0, then the value of n is, , m 0, , (a) 15, 134. Let 0  x , , (b) 6, , , 4, , (c) 1, , (d) 0, , . Then sec 2 x  tan 2 x =, , , , (a) tan  x  , 4, , , , , (b) tan   x , 4, , , [IIT 1994], , , , (c) tan  x  , 4, , , , , (d) tan 2  x  , 4, 

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42 Trigonometrical Ratios, Functions and, Identities, 135. If x is A.M. of tan, , 9, , and tan, , 7, 5, , and y is A.M. of tan, and tan, , then, 18, 18, 9, , (b) x  y, , (a) x  y, 136. If cos 4  sec 2  ,, (a) A.P., 137. Let fn ( )  tan, , , , (c) 2x = y, , 1, 1, and sin 4  cosec 2 are in A.P. , then cos 8  sec 6  , and sin 8  cosec 6 are in, 2, 2, , (b) G.P., , , 2, ,  , (a) f 2    1,  16 , , (d) x = 2y, , (c) H.P., , (d) None of these, , (1  sec  )(1  sec 2 )(1  sec 4 ) ……. (1  sec 2 n  ) . Then, , [IIT Screening 1999; 2001], ,   , (c) f4 , 1,  64 , ,  , (b) f3    1,  32 , , (d) All of these, , 138. If A, B, C, D are the smallest positive angles in ascending order of magnitude which have their sines equal to, A, B, C, D, the positive quantity k, then the value of 4 sin  3 sin  2 sin  sin, is equal to, 2, 2, 2, 2, (a) 2 1  k, , (b) 2 1  k, , (c) 2 k, , (d) None of these, ,    , 139. If ,  are different values of x satisfying a cos x  b sin x  c, then tan , =,  2 , , [Orissa JEE 2003;, , EAMCET 1986], , (a) a  b, , (b) a  b, , (c), , b, a, , (d), , a, b, , Maximum and Minimum value, , Basic Level, 140. The maximum value of a cos x  b sin x is, (a) a  b, , [MNR 1991; MP PET 1999], , (b) a  b, , (c) | a |  | b |, , (d) (a 2  b 2 )1 / 2, , 141. The minimum value of cos   sin  is, (a) 0, , [MNR 1976], , (b)  2, , (c) 1/2, , (d), , 2, , 142. The minimum value of 3 cos x  4 sin x  8 is, (a) 5, , (b) 9, , 143. If  is an acute angle and sin , (a) 6  p  8, , [UPSEAT 1991], , (c) 2, , (d) 3, , p 6, , then p must satisfy, 8p, , (c) 3  p  4, , (b) 6  p  7, , (d) 4  p  7, , Advance Level, , 144. Maximum value of cos 2 x  cos 2 y  cos 2 z is, (a) 0, , (b) 1, , , 145. Let n be a positive integer such that sin n, 2, , (c) 3, , ,   cos  n, , 2, , n, , , then, , 2, , , (d) 2

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Trigonometrical Ratios, Functions and Identities, (a) 6  n  8,  , 146. If    0, , then,  2, , x2  x , , (a) 2 tan , 147. The, , tan 2 , x2  x, , is always greater than or equal to, , (b) 1, , maximum, , value, , [IIT Screening 2003], , (d) sec 2 , , (c) 2, , (cos 1 ).(cos  2 ).....(cos  n ) ,, , of, , under, , the, , restrictions,, , 0   1 ,  2 ......  n , , (cot 1 ).(cot  2 )........( cot  n )  1 is, (a), , 1, 2n, , (b), , 2, , , 2, , and, , [IIT Screening 2001], , 1, , (c), , 2n, , 1, 2n, , (d) 1, , 148. Let f ( ) = sin  (sin   sin 3 ) . Then, (a), , 43, , (d) 4  n  8, , (c) 4  n  8, , (b) 4  n  8, , [IIT Screening 2000], , f ( )  0 only when   0, , (b) f ( )  0 only when   0, , (c) f ( )  0 for all real , , (d) None of these, , 149. The minimum value of 2sin x  2cos x is, , , (a) 1, , (b) 2, , (c) 2, , 1, , 1, , 1, , 2, , 2, , (d) 2, , Conditional Trigonometrical Identities, , Basic Level, , 150. If A  B  C   , then, (a) 0, 151., , tan A  tan B  tan C, , tan A. tan B. tan C, , [EAMCET 1989], , (b) 2, , (c) 1, , (d) –1, , If A  B  C   and cos A  cos B cos C, then tan B tan C is equal to, (a), , 1, 2, , (b) 2, , (c) 1, , [AMU 2001], , (d) , , 1, 2, , Advance Level, , 152. If        , then sin 2   sin 2   sin 2  , (a) 2 sin  sin  cos , 153. If A  B  C , (a) 0, , (b) 2 cos  cos  cos , , [IIT 1980], , (c) 2 sin  sin  sin , , (d) None of these, , 3, , then cos 2 A  cos 2B  cos 2C  4 sin A sin B sin C , 2, , (b) 1, , (c) 2, , [EAMCET 2003; 1989], , (d) 3, , 154. If A, B, C are the angles of a triangle, then sin 2 A  sin 2 B  sin 2 C  2 cos A cos B cos C =, (a) 1, , (b) 2, , (c) 3, , [Karnataka CET 1989], , (d) 4

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44 Trigonometrical Ratios, Functions and, Identities, 155. If    , , , 2, , and      , then tan  equals, , (a) 2(tan   tan  ), , [IIT Screening 2001], , (c) tan   2 tan , , (b) tan   tan , , 156. Let A, B and C are the angles of a plain triangle and tan, (a) 7/9, , (b) 2/9, , (d) 2 tan   tan , , A 1, B 2, C,  , tan  . Then tan, is equal to, 2, 3, 2, 3, 2, , (c) 1/3, , (d) 2/3, , ***, , 44, , Trigonometrical, Identities, , Ratios,, , Functions, , Assignment (Basic & Advance Level), , and, , 1, , 2, , 3, , 4, , 5, , 6, , 7, , 8, , 9, , 10, , 11, , 12, , 13, , 14, , 15, , 16, , 17, , 18, , 19, , 20, , b, , b, , b, , d, , b, , d, , a, , c, , a, , b, , a, , b, , c, , a, , b, , c, , c, , c, , c, , c, , 21, , 22, , 23, , 24, , 25, , 26, , 27, , 28, , 29, , 30, , 31, , 32, , 33, , 34, , 35, , 36, , 37, , 38, , 39, , 40, , b, , d, , a, , a, , b, , a, , d, , a, , c, , b,d, , d, , c, , c, , b, , b, , a, , a, , d, , d, , d, , 41, , 42, , 43, , 44, , 45, , 46, , 47, , 48, , 49, , 50, , 51, , 52, , 53, , 54, , 55, , 56, , 57, , 58, , 59, , 60, , a, , a, , c, , c, , b, , b, , a, , d, , a, , c, , a, , c, , a, , c, , a, , d, , b, , c, , d, , d, , 61, , 62, , 63, , 64, , 65, , 66, , 67, , 68, , 69, , 70, , 71, , 72, , 73, , 74, , 75, , 76, , 77, , 78, , 79, , 80, , b, , a, , d, , c, , a, , a, , b, , b, , a, , a, , a, , a, , c, , d, , b, , d, , c, , b, , a, , c

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Trigonometrical Ratios, Functions and Identities, , 45, 81, , 82, , 83, , 84, , 85, , 86, , 87, , 88, , 89, , 90, , 91, , 92, , 93, , 94, , 95, , 96, , 97, , 98, , 99, , 100, , b, , a, , a, , a, , b, , d, , d, , d, , c, , a, , b, , a, , d, , d, , d, , a, , a, , a,, , d, , c, , b, 101, , 102, , 103, , 104, , 105, , 106, , 107, , 108, , 109, , 110, , 111, , 112, , 113, , 114, , 115, , 116, , 117, , 118, , 119, , 120, , a, , b, , b, , c, , c, , c, , a, , b, , b, , c, , d, , b, , d, , c, , b, , c, , b, , c, , b, , b, , 121, , 122, , 123, , 124, , 125, , 126, , 127, , 128, , 129, , 130, , 131, , 132, , 133, , 134, , 135, , 136, , 137, , 138, , 139, , 140, , c, , a, , a, , c, , b, , a, , b, , c, , a, , a, , a, c, , b, , b, , b, , c, , a, , d, , b, , c, , d, , 141, , 142, , 143, , 144, , 145, , 146, , 147, , 148, , 149, , 150, , 151, , 152, , 153, , 154, , 155, , 156, , b, , d, , b, , d, , b, , a, , a, , c, , d, , c, , b, , a, , b, , b, , c, , a