Question

If $y=\frac{1}{2}\log \left(\frac{1-\cos 2x}{1+\cos 2x}\right)$then prove that $\frac{dy}{dx}=2cosec2x$.

Answer 1

$y=\frac{1}{2}\left[\frac{1-\cos 2x}{1+\cos 2x}\right]PT\frac{dy}{dx}=2cosec2x$

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• $\frac{1-\cos 2x}{1+\cos 2x}={\tan }^{2}x$

$\Rightarrow y=\frac{1}{2}\log \left({\tan }^{2}x\right)$

$\frac{dy}{dx}=\frac{1}{\sin x\cos x}$

Multiply and divide by $2$

$\frac{dy}{dx}=\frac{2}{2\sin x\cos x}$

$\frac{dy}{dx}=\frac{2}{\sin 2x}=2cosec2x$ proved.