Question

If $Y={\cot }^{-1}\left[\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right]$. Prove that $\frac{dy}{dx}$ is independent of $x$.

Answer 1

Multiplying and dividing by $(\sqrt{1+\sin x}+\sqrt{1-\sin x})$ inside ${\cot }^{-1}$:-

$y={\cot }^{-1}\left[\frac{(1+\sin x)+(1-\sin x)+2\sqrt{1-{\sin }^{2}x}}{(1+\sin x)-(1-\sin x)}\right]$

$={\cot }^{-1}\left[\frac{2+2\cos x}{2\sin x}\right]$

$={\cot }^{-1}\left(\frac{1+\cos x}{\sin x}\right)$ As $\cos x=2{\cos }^{2}x/2-1$

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

$\Rightarrow y={\cot }^{-1}\left(\cot x/2\right)$

$\Rightarrow y=\frac{x}{2}\Rightarrow \frac{dy}{dx}=\frac{1}{2}$