Mathematics - Inverse Trigonometric Functions Question with Solution | TestHub

$$\begin{array}{rlrlr}& {\left({\tan }^{-1}x\right)}^2+{\left({\cot }^{-1}x\right)}^2={\left({\tan }^{-1}x+{\cot }^{-1}x\right)}^2-2{\tan }^{-1}x\left(\frac{\pi }{2}-{\tan }^{-1}x\right)& & & \end{array}$$

$$=\frac{{\pi }^2}{4}-\pi {\tan }^{-1}x+2{\left({\tan }^{-1}x\right)}^2=\frac{5{\pi }^2}{8}$$

$$\therefore {\tan }^{-1}x=\frac{2\pi }{3},-\frac{\pi }{4}{\tan }^{-1}x=-\frac{\pi }{4}\left\{{\tan }^{-1}x\ne \frac{2\pi }{3}\right\}$$

$$\therefore x=-1isthesolution$$