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1. (i) Prove that: sin®e , cos*e=1- 3sin’ecos’e (ii) Prove (1+cotA-cosecA)(1+tanA+secA) = 2, 2. If sin A+ cos A = x , prove that sin® A +cos® A= [4 - 3(x? - 1)|/4, , 3. if cosec A- sin Aza*, sec A - cos A=b * prove that a*b’(a*+b*) = 1, , 4. If sec A+ tan A =p, then find the value of sec A - tan A, , 5. Prove that : (tand + secO — 1).(tan0+ 1+ secd) ==“, , 1-sing, , we 4, 6. if sind = == and d >0 find the value of cos @ and tan 6, , 7. If sin A +sin’A+sin°A=1, prove that cos"A-4cos“A+8cos’A= 4., , m7+n?_ -1 cot A+tan A, , 8. ifm =cos A-sin A and n = cos A + sin A, show that ——> = + SecA. Cosec A = :, , 9. Prove that (sin@ - cos® + 1)/ (sin6 + cos6 — 1) = 1/ (sec@ — tan8), using the identity sec’ @ = 1 + tan’9, 10, Write all other trignometrical identities in form of cot A?, , 11. \f ZA and 2B are acute angles such that cos A = cos B, then show that 2A = 2B., , 12, Express sin 67° + cos 75° in terms of trigonometric ratios of angles between 0° and 45°,, , 13. Prove that cos 4 - sin 6 =v 2 sin 0, if sin 6 + cos @ = V2 cos 0, , 14. Prove that (tan A+ sec A- 1)/ (tan A- secA+1)=secA+ tanA, , 15. A, B, and C are the interior angles of A ABC show that sin( B + C }/2 = cos A/2, , 16. In AABC, if sin (A +B -C) = V3/2 and cos(B + C — A) =1/v2, find A, B and C., , 16. In A OPQ, right-angled at P,OP = 7cm and OO — PQ = 1cm. Determine the values of sin Q and cosQ., 18. Iftan A+ sin A=m and tan A - sin A= n show that m?=n* =4, , 19, Ifx=psecA+qtanA and y=p tan A +q sec A prove that x’ — y’ = p’-q*, , 20. Prove that : (i) sec*e - sec*9 = tan‘ 0 + tan*0 (ii) Show that (1-sinA+cosA) 2=2(1+cosA)(1-sinA), 21. Given 2cos36 = ¥'3 . find the value of 6., , 22. When is an equation called ‘an identity’. Prove the trigonometric identity 1+ tan? A = sec? A, , , , , , a, 23. Ifsech + tan @ =p, show that == cosec@ = 1, , ign 2, 24. Ifx=tan A + sin Aand y=tan A-sin A, prove that: (=) - & =1, , tan A+ tanB, 1—tanA tanB, , 25. If cose (A+B) = 1 and sec(A - B) = 2, evaluate : (i) sin A cos B +cos A sin B. (ii), 26. (i) In aA ABC, write tan “**? in terms of angle C. (ii) If sind - cos 0 = 0,0" < 6 < 90°, find the value of 0., , 27. If cot 8 = 3x- —— . then show that cot + cosecd = 6x or , sin A i+cus A sin A, , 2 1—cosA\ _, 28. Prove that : (“4+ a) ae os) = 4 cosec A.cota, , 29. Given that cos (A- B) = cos A. cos B+sin A. sin B, find the value of cos15° in two ways., (i) Taking A = 60°, B = 45° and (ii) Taking A= 45° and B = 30°, 30.(i) If sec 0 - tan 0 = x show that secd => (= +x) and tano +(=— x), , (ii) If sec 0 + tan @ =x show that seco =+(++2x) and tano+(+— x), 31. Simplify; Seas" 32. Show that (sin A + cosec A)’ + (cos A+ sec A)’ =7 + tan’ A+ cot? A, aera