2. If tan + If tan 3+ sin²x = tan (...
Question

# 2. If tan + If tan 3+ sin²x = tan ( + ) then prove that siny = sinx 73 sin? Bit + inx then prove that sin NIX

JEE/Engineering Exams
Maths
Solution
59
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( tan frac{pi}{4}+tan frac{y}{2}=left[frac{tan pi / 4+tan ^{x}(2)}{sin tan frac{pi}{4} tan x}right] ) ( tan pi / 4 tan ^{4} n ) ( Rightarrow frac{1+tan y / 2}{1-tan y / 2}=left[frac{1+tan frac{x}{2}}{1-tan x / 2}right]^{3} ) ( Rightarrowleft(frac{cos y / 2+sin y / 2}{cos y / 2-sin y / 2}right)^{2}=left[frac{cos x / 2+sin x / 2}{cos x / 2-sin x / 2}right]^{3} ) ( Rightarrowleft(frac{cos y / 2+sin y / 2}{cos y(2-sin y / 2)}=left[frac{cos x / 2+sin x / 2}{cos x / 2-sin x / 2}right]^{2}right]^{3} ) an ( ^{2} y / 2+sin ^{2} y / 2+2 cos y / 2 sin y / 2 ) ( cos ^{2} y / 2+sin ^{2} y / 2-2 cos y / 2 sin y / 2 ) ( left.=frac{left[log ^{2} x / 2+sin ^{2} x / 2+2 cos x / 2 sin ^{4} 12right.}{cos ^{2} x_{12}+sin ^{2} x / 2-2 cos x / 2 sin x / 2}right] ) ( Rightarrow frac{1+sin y^{2}}{1-sin y}=left[frac{1+sin x}{1-sin x}right]^{3} ) 2) ( frac{1+sin y+1-sin y}{1+sin y-1+sin y}=frac{11+sin x^{8}+[1-sin x]^{3}}{(1+sin x)^{3}-(1-sin x)^{3}} ) ( Rightarrow frac{2}{2 sin y}=frac{(2)(1+sin x)^{2}-(1+sin x)(1-sin x)}{2 sin xleft((1+sin x)^{2}+(1+sin x)(1-sin x)+(1-sin x)right.} ) ( Rightarrow frac{1}{sin y}=frac{1}{sin x}left(frac{3 sin ^{2} x+1}{3+sin ^{2} x}right) ) ( Rightarrow sin y=sin frac{1left(3+sin ^{2} xright)}{1+3 sin ^{2} x} ) ) ( frac{sin y}{sin x}=frac{3+sin ^{2} x}{3 sin ^{2} x+1}= )