Question

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

Answer 1

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

$\Rightarrow \cos ({\tan }^{-1}x+{\cot }^{-1}\sqrt{3})=\cos \frac{\pi }{2}$ $[\because \cos \frac{\pi }{2}=0]$

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

We know that,

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

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

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

$\Rightarrow -{\cot }^{-1}x=-{\cot }^{-1}\sqrt{3}$

$\Rightarrow {\cot }^{-1}x={\cot }^{-1}\sqrt{3}$

$\therefore x=\sqrt{3}$