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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