Question

$5{\tan }^{-1}x+3{\cot }^{-1}x=2\mathrm{\pi }$

Answer 1

$$\begin{gathered} 5{\tan }^{-1}x+3{\cot }^{-1}x=2\mathrm{\pi } \\ \Rightarrow 5{\tan }^{-1}x+3\left(\frac{\mathrm{\pi }}{2}-{\tan }^{-1}x\right)=2\mathrm{\pi }\left[\because {\cot }^{-1}x=\frac{\mathrm{\pi }}{2}-{\tan }^{-1}x\right] \\ \Rightarrow 5{\tan }^{-1}x+\frac{3\mathrm{\pi }}{2}-3{\tan }^{-1}x=2\mathrm{\pi } \\ \Rightarrow 2{\tan }^{-1}x=\frac{\mathrm{\pi }}{2} \\ \Rightarrow {\tan }^{-1}x=\frac{\mathrm{\pi }}{4} \\ \Rightarrow \mathrm{x}=\tan \frac{\mathrm{\pi }}{4}=1 \end{gathered}$$