Question

The value of $\cos [{{\tan }^{-1}}\\{\sin ({{\cot }^{-1}}x)\\}]$ is

• A. $\sqrt{\frac{{{x}^{2}}+1}{{{x}^{2}}-1}}$
• B. $\sqrt{\frac{1-{{x}^{2}}}{{{x}^{2}}+2}}$
• C. $\sqrt{\frac{1-{{x}^{2}}}{1+{{x}^{2}}}}$
• D. $\sqrt{\frac{{{x}^{2}}+1}{{{x}^{2}}+2}}$

Answer 1

Answer: (D)

$\sqrt{\frac{{{x}^{2}}+1}{{{x}^{2}}+2}}$

Answer 2

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