Question

A value of $x$ satisfying the equation $\sin \left[{\cot }^{-1}\right(1+x\left)\right]=\cos \left[{\tan }^{-1}x\right]$, is.

• A. $-\frac{1}{2}$
• B. $\frac{1}{2}$
• C. $-1$
• D. $0$

Answer 1

The correct option is A $-\frac{1}{2}$

From first figure,

$\sin \left({\cot }^{-1}\right(1+x\left)\right)$ $=\frac{1}{\sqrt{1+1+x^2+2x}}$ $=\frac{1}{\sqrt{x^2+2x+2}}$

From second figure,
$\cos \left({\tan }^{-1}x\right)$ $=\frac{1}{\sqrt{1+x^2}}$
$\sin \left({\cot }^{-1}\right(1+x\left)\right)=\cos \left({\tan }^{-1}\right(x\left)\right)$
$\Rightarrow x^2+2x+2=1+x^2$
$\Rightarrow 2x=-1$

$\Rightarrow x=-\frac{1}{2}$