Mathematics - Inverse Trigonometric Functions Question with Solution | TestHub

${\cot }^{-1}(\sqrt{\cos \alpha })-{\tan }^{-1}(\sqrt{\cos \alpha })=x$ ${\tan }^{-1}\left(\frac{1}{\sqrt{\cos \alpha }}\right)-{\tan }^{-1}(\sqrt{\cos \alpha })=x\Rightarrow {\tan }^{-1}\frac{\frac{1}{\sqrt{\cos \alpha }}-\sqrt{\cos \alpha }}{1+\frac{1}{\sqrt{\cos \alpha }}\cdot \sqrt{\cos \alpha }}=x$ $\Rightarrow {\tan }^{-1}\frac{1-\cos \alpha }{2\sqrt{\cos \alpha }}=x\Rightarrow \tan x=\frac{1-\cos \alpha }{2\sqrt{\cos \alpha }}$ OR $\cot x=\frac{2\sqrt{\cos \alpha }}{1-\cos \alpha }$ or $\mathrm{cosec}x=\frac{1+\cos \alpha }{1-\cos \alpha }$ $\sin x=\frac{1-\cos \alpha }{1+\cos \alpha }=\frac{1-\left(1-2{\sin }^2\alpha /2\right)}{1+2{\cos }^2\alpha /2-1}$ or $\sin x={\tan }^2\frac{\alpha }{2}$