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Q.
$f(x)=\cos \left(\cot ^{-1}\left(\sin \left(\tan ^{-1} x\right)\right)\right)$ then number of solution(s) of the equation $\sqrt{3} f ( x )-1=0$ is (are)
Inverse Trigonometric Functions
Solution:
$f ( x )=\frac{1}{\sqrt{3}} $
$\Rightarrow \cos \left(\cot ^{-1}\left(\sin \left(\tan ^{-1} x \right)\right)\right)=\frac{1}{\sqrt{3}} $
$\Rightarrow \cot ^{-1}\left(\sin \left(\tan ^{-1}\right)\right)=\cos ^{-1}\left(\frac{1}{\sqrt{3}}\right)$
$\Rightarrow \cot ^{-1}\left(\sin \left(\tan ^{-1} x \right)\right)=\cot ^{-1}\left(\frac{1}{\sqrt{2}}\right)$
$\Rightarrow \sin \left(\tan ^{-1} x \right)=\left(\frac{1}{\sqrt{2}}\right)$
$\Rightarrow \tan ^{-1} x =\frac{\pi}{4} \Rightarrow x =1$