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3. DOMAIN AND RANGE OF TRIGONOMETRICAL FUNCTIONS, Domain, Range, sin x, R, [-1, 1], COS X, R, [-1,1], R- {(2 n+ 1) T/2:ne Z} (-0, 00) = R, (-00, - 1]U[1, ), R-(2 n+1) T/2:neZ} (-, – 1]U[1, 0), (- 00, 0) = R, Thus, | sin x| <1,| cos x| <1,sec x 21 or, sec x <-1 and,, tan x, %3D, cosec x R- ((n n:n eZ}, sec x, cot x, R-n T:neZ}, %3D, cosec x 21 or, cosec x 3-1., 4. SUM AND DIFFERENCE FORMULAE, 1. sin (x +y), sin x cos y + cos x sin y, %3D, 2. sin (x - y), sin x cos y, cos x sin y, |, 3. cos (x + у), COS X COS y, sin x sin y, |, 4. cos (x -y), = COS X Cos y + sin x sin y, + tan y, tan, 5. tan (x + y) =, TC, TC, where x nn+-,y#nT +, %3D, 1-tan x tan y, TC, tan x - tan y, and, x t y # nT +, 6. tan (x-y), 1+ tan x tan y, cot x cot y-1, 7. cot (x + y) =, %3D, cot x + cot y, wherex # nT, Y# NT, cot x cot y+1, and,x ty #nT, cot (x - y) =, %3D, cot y - cot x, %3D, 8., sin (x + y) sin (x-y) = sinx-sin y = cosx-cos² y, cos(x + y) cos(x - y) = cos x- siny = cos y - sin?x, %3D, 9., 2 tan x, 10. sin 2x = 2 sin x cos x =, 1 + tan-, 11., cos 2x = cos- x - sin2x, %3D

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2 cos? x -1, COs 2x, %3D, cos 2x = 1 - 2 sin2 x, %3D, 1 - tan?, cos 2x, %3D, 1 + tan2x, 1 + cos 2x = 2 cos x, 1- cos 2x = 2 sin2 x, 12., COS, 1- cos 2x, 1 + cos 2x, or,, = sin x, = cos x,, %3D, 2, 2 tan x, TC, where x# (2n + 1), 4, 13., tan 2x =, %3D, 1 tan x, 2, 1- cos X, 14., = tan, where x # 2n T, sin x, 1 + COS x, 15., where x # (2n+ 1) T, = cot -, 2, sin x, 1 - cos x, 16., tan, where x # (2n + 1) t, 1 + cosx, 1 + Cos X, = cot, 2', 2 x, 17., where x # 2 n n, %3D, 1- cos x, 18. sin 3x = 3 sin x -4 sin, %3D, 19. cos 3x = 4 cos° x - 3 cosx, %3D, 3, 3 tan x- tan° x, 20., tan 3x, %3D, 1-3 tan x, |, sin 2" x, 21. cos x cos 2x cos 24x... cos 2"-1,, 2 sin x, 1, 22., sin x sin (T/3 - x) sin (T/3 + x) == sin 3x, 4, 23., cos x cos (T/3-x) cos (1/3 +x) = – cos 3x, 1, 4, 24., tan x tan (T/3 – x) tan (T/3 +x) = tan 3x

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5. SUM AND DIFFERENCE INTO PRODUCTS, x+ y, 1. sin x + sin y =2 sin, 2, X-y, COS, 2. sin x- sin y =2 sin, x-y, x+y, COS, x+ y, x-y, COS, 3., cos x + cos y = 2 cos, x-y, sin, 4. cos x- cos y =- 2 sin, sin (x + y), tan x + tan y:, TC, COs x COs y, wherex, y # n T+,ne Z, sin (x – y), 6., tan x - tan y =, %3D, COs X Cos y, 5.

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sin (x+ y), 7. cot x+ cot y =, %D, sin x sin y, where x, y #N 1, Ne Z, sin (y-x), sin x sin y, 8., cot x - cot y, %3D, 6. PRODUCT INTO SUM OR DIFFERENCE, 2 sin x cos y = sin (x + y) + sin (x-y), 2 cos x sin y = sin (x + y) – sin (x - y), 1., %3D, 2., 2 cos x cos y = cos (x + y) + cos (x -y), 2 sin x sin y = cos (x-y) - cos (x + y), 3., 4., %3D