Question

Prove that ${\tan }^{-1}\left(\cos ecx-\cot x\right)=\frac{x}{2}.$

Answer 1

${\tan }^{-1}({\csc }^{-1}-\cot x)=x/2$

$\Rightarrow {\tan }^{-1}(\frac{1}{\sin x}-\frac{\cos x}{\sin x})=x/2$

$\Rightarrow {\tan }^{-1}\left(\frac{2{\sin }^{2}x/2}{2\sin x/2\cos x/2}\right)$

$\Rightarrow {\tan }^{-1}\left({\tan }^{-1}x/2\right)$

$\Rightarrow x/2$