Question

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

Answer 1

$$\begin{gathered} \cos \left({\tan }^{-1}x+{\cot }^{-1}\sqrt{3}\right)=0 \\ \Rightarrow \cos \left({\tan }^{-1}x+{\cot }^{-1}\sqrt{3}\right)=\cos \left(\frac{\mathrm{\pi }}{2}\right) \\ \Rightarrow {\tan }^{-1}x+{\cot }^{-1}\sqrt{3}=\frac{\mathrm{\pi }}{2} \\ \Rightarrow x=\sqrt{3}\left[\because {\tan }^{-1}y+{\cot }^{-1}y=\frac{\mathrm{\pi }}{2}\right] \end{gathered}$$