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Question:

If $\cos \left(\tan ^{-1} x+\cot ^{-1} \sqrt{3}\right)=0$, find the value of $x$.

Solution:

$\cos \left(\tan ^{-1} x+\cot ^{-1} \sqrt{3}\right)=0$

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

$\Rightarrow \tan ^{-1} x+\cot ^{-1} \sqrt{3}=\frac{\pi}{2}$

$\Rightarrow x=\sqrt{3} \quad\left[\because \tan ^{-1} y+\cot ^{-1} y=\frac{\pi}{2}\right]$

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