Find the derivative of the following function: $\operatorname{cosec} x \cot x$.

(N/A) Let $f(x) = \operatorname{cosec} x \cot x$.
Using the product rule for differentiation,$\frac{d}{dx}[u(x)v(x)] = u(x)v'(x) + v(x)u'(x)$.
Here,$u(x) = \operatorname{cosec} x$ and $v(x) = \cot x$.
We know that $\frac{d}{dx}(\operatorname{cosec} x) = -\operatorname{cosec} x \cot x$ and $\frac{d}{dx}(\cot x) = -\operatorname{cosec}^2 x$.
Applying the product rule:
$f'(x) = \operatorname{cosec} x \frac{d}{dx}(\cot x) + \cot x \frac{d}{dx}(\operatorname{cosec} x)$
$f'(x) = \operatorname{cosec} x (-\operatorname{cosec}^2 x) + \cot x (-\operatorname{cosec} x \cot x)$
$f'(x) = -\operatorname{cosec}^3 x - \operatorname{cosec} x \cot^2 x$
$f'(x) = -\operatorname{cosec} x (\operatorname{cosec}^2 x + \cot^2 x)$.