solve this
Question:

$y=\sqrt{\frac{1+\cos 2 x}{1-\cos 2 x}}$, find $\frac{d y}{d x}$

 

Solution:

$y=\sqrt{\frac{1+\cos 2 x}{1-\cos 2 x}}$

Formula:

Using double angle formula:

$\cos 2 x=2 \cos ^{2} x-1$

$=1-2 \sin ^{2} x$

$\therefore 1+\cos 2 x=2 \cos ^{2} x$

$1-\cos 2 x=2 \sin ^{2} x$

$\therefore y=\sqrt{\frac{2 \cos ^{2} x}{2 \sin x^{2} x}}$

$=\sqrt{\cot ^{2} x}$

$=\cot x$

Differentiating $y$ with respect to $x$

$\frac{d y}{d x}=\frac{d}{d x}(\cot x)$

$=-\operatorname{cosec}^{2} x$

 

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