Question

( x=tan ^{2} theta, ) we get
soution Putting
[
begin{aligned}
R H S &=frac{1}{2} cos ^{-1}left(frac{1-x}{1+x}right)
&=frac{1}{2} cos ^{-1}left(frac{1-tan ^{2} theta}{1+tan ^{2} theta}right)
&=frac{1}{2} cos ^{-1}(cos 2 theta)
=&left(frac{1}{2} times 2 thetaright)=theta=tan ^{-1} sqrt{x}=mathrm{LHS}
&left[because x=tan ^{2} theta Rightarrow tan theta=sqrt{x} Rightarrow theta=tan ^{-1} sqrt{x}right]
text { Hence, } tan ^{-1} sqrt{x}=frac{1}{2} cos ^{-1}left(frac{1-x}{1+x}right) &
end{aligned}
]

# Prove that tan " Var = LC05-1 (1-x),x € [0, 1] 1 + x

Solution