Question

If ${\sin }^{-1}x+{\cot }^{-1}\left(\frac{1}{2}\right)=\frac{\pi }{2},$ then $x$ is equal to

• A. $0$
• B. $\frac{1}{2}$
• C. $\frac{1}{\sqrt{5}}$
• D. $\frac{2}{\sqrt{5}}$

Answer 1

The correct option is C $\frac{1}{\sqrt{5}}$

${\cot }^{-1}\left(\frac{1}{2}\right)={\cos }^{-1}\left(\frac{1}{\sqrt{5}}\right)$
$\text{Given}{\sin }^{-1}x+{\cot }^{-1}\left(\frac{1}{2}\right)=\frac{\pi }{2}$
$\Rightarrow {\sin }^{-1}x+{\cos }^{-1}\left(\frac{1}{\sqrt{5}}\right)=\frac{\pi }{2}$
$\text{Thus,}x=\frac{1}{\sqrt{5}}$