Question

$f(x)=\cos \left({\cot }^{-1}\left(\sin \left({\tan }^{-1}x\right)\right)\right)$ then number of solution(s) of the equation $\sqrt{3}f(x)-1=0$ is (are)

• A. 0
• B. 1
• C. 2
• D. infinite

Answer 1

Answer: (B)

$f(x)=\frac{1}{\sqrt{3}}$
$\Rightarrow \cos \left({\cot }^{-1}\left(\sin \left({\tan }^{-1}x\right)\right)\right)=\frac{1}{\sqrt{3}}$
$\Rightarrow {\cot }^{-1}\left(\sin \left({\tan }^{-1}\right)\right)={\cos }^{-1}\left(\frac{1}{\sqrt{3}}\right)$
$\Rightarrow {\cot }^{-1}\left(\sin \left({\tan }^{-1}x\right)\right)={\cot }^{-1}\left(\frac{1}{\sqrt{2}}\right)$
$\Rightarrow \sin \left({\tan }^{-1}x\right)=\left(\frac{1}{\sqrt{2}}\right)$
$\Rightarrow {\tan }^{-1}x=\frac{\pi }{4}\Rightarrow x=1$