sin e Then sin , x = θ, we havesin.' ( 2xv1-x, )sin'( 2 sin θ ./l-si

1.	sin e Then sin , x = θ, we havesin.' ( 2xv1-x, )sin'( 2 sin θ ./l-sin , θ )- sin ' (2sine cos8)- sin (sin20)- 22 sin xcos θ, then proceeding as above, we get. sını (2xJ1-r )=22-+tan-111=tan-1-4how thattan:roperty 5 (i). we have2-i2 11221520.tan+ tantan= tan2 11-Go sin/,-π < x <3t inthe simplestcos xpress tarnrite2cos-sin2sin xcos +sin2 -2sin cos
Answer» let y = sin x/(1-cos x) we need to find t such that tan t = y ..1 we know sin x = 2 sin x/2 cos x/2cos x = 1- 2 sin^2 x/2so 1-cos x = 2 sin ^2 x/2 now y = sin x/(1-cos x) = (2 sin x/2 cos x/2)/(2 sin ^2 x/2) so y = cot x/2 = tan (pi/2 - x/2) as cot a = tan (pi/2 -a) using 1 t = pi/2 - x/2 orarctan ( sin (x)/(1-cos x)) = pi/2 - x/2

sin e Then sin , x = θ, we havesin.' ( 2xv1-x, )sin'( 2 sin θ ./l-sin , θ )- sin ' (2sine cos8)- sin (sin20)- 22 sin xcos θ, then proceeding as above, we get. sını (2xJ1-r )=22-+tan-111=tan-1-4how thattan:roperty 5 (i). we have2-i2 11221520.tan+ tantan= tan2 11-Go sin/,-π < x <3t inthe simplestcos xpress tarnrite2cos-sin2sin xcos +sin2 -2sin cos

Answer»

let y = sin x/(1-cos x)

we need to find t such that

tan t = y ..1

we know sin x = 2 sin x/2 cos x/2cos x = 1- 2 sin^2 x/2so 1-cos x = 2 sin ^2 x/2

now y = sin x/(1-cos x) = (2 sin x/2 cos x/2)/(2 sin ^2 x/2)

so y = cot x/2 = tan (pi/2 - x/2) as cot a = tan (pi/2 -a)

using 1 t = pi/2 - x/2

orarctan ( sin (x)/(1-cos x)) = pi/2 - x/2