Question

SOLUTION Putting ( x=tan theta ) and ( y=tan phi ) we get
[
begin{aligned}
L H S &=tan left{frac{1}{2} sin ^{-1}left(frac{2 x}{1-x^{2}}right)+frac{1}{2} cos ^{-1}left(frac{1-y^{2}}{1+y^{2}}right)right}
&=tan left{frac{1}{2} sin ^{-1}left(frac{2 tan theta}{1-tan ^{2} theta}right)+frac{1}{2} cos ^{-1}left(frac{1-tan ^{2} phi}{1+tan ^{2} phi}right)right}
&=tan left{frac{1}{2} sin ^{-1}left(sin 2 theta+frac{1}{2} cos ^{-1}(cos 2 phi)right}right.
&=tan left{left(frac{1}{2} times 2 thetaright)+left(frac{1}{2} times 2 phiright)right}
&=tan (theta+phi)=frac{tan theta+tan phi}{1-tan theta tan phi}=frac{(x+y)}{(1-x y)}=R H S
end{aligned}
]

# + - cos + EXAMPLE 40 Prove that tan (1-x) wherel x <1, y> 0 and xy <1. 2 (1+y? 1 1- xy)" [CBSE 2016

Solution