Mathematics - Inverse Trigonometric Functions Question with Solution | TestHub

$2{\tan }^{-1}\left(\mathrm{cosec}\left({\tan }^{-1}x\right)-\cot \left({\tan }^{-1}x\right)\right)=2{\tan }^{-1}\frac{1}{3}+{\tan }^{-1}\frac{1}{7}$

$$\begin{array}{rl}\mathrm{LHS}& =2{\tan }^{-1}\left(\mathrm{cosec}\left({\sin }^{-1}\left(\frac{x}{\sqrt{1+x^2}}\right)-\cot \left({\cot }^{-1}\frac{1}{x}\right)\right)\right)\\ & =2{\tan }^{-1}\left(\frac{\sqrt{1+x^2}}{x}-\frac{1}{x}\right)\\ & ={\tan }^{-1}\left(\frac{2\left(\sqrt{1+x^2}-1\right)x}{x^2-{\left(\sqrt{1+x^2}-1\right)}^2}\right)={\tan }^{-1}x\\ \mathrm{RHS}& ={\tan }^{-1}\left(\frac{3}{4}\right)+{\tan }^{-1}\frac{1}{7}={\tan }^{-1}1\\ \Rightarrow x& =1\end{array}$$