Question

The value of $x$ for which $\sin \left(\cot ^{-1}(1+x)\right)=\cos \left(\tan ^{-1} x\right)$ is

(A) $1 / 2$

(B) 1

(C) 0

(D) $-1 / 2$

Answer 1

SOLUTION :  $ \sin \cot ^{-1}(1+x)=\cos \left(\tan ^{-1} x\right)$

If $\alpha=\cot ^{-1}(1+x) \quad$ and $\beta=\tan ^{-1} x$

Then

$\frac{1}{\sqrt{x^2+2 x+2}}=\frac{1}{\sqrt{1+x^2}} \Rightarrow x=-1 / 2$