## Task Condition

Find the derivative.

$$
y=\frac{\cos x}{\sin ^{2} x}-2 \cos x-3 \ln \left(\tan \frac{x}{2}\right)
$$
## Solution

$y^{\prime}=\left(\frac{\cos x}{\sin ^{2} x}-2 \cos x-3 \ln \left(\operatorname{tg} \frac{x}{2}\right)\right)^{\prime}=$

$=\frac{-\sin x \cdot \sin ^{2} x-\cos x \cdot 2 \sin x \cdot \cos x}{\sin ^{4} x}+2 \sin x-3 \cdot \frac{1}{\operatorname{tg} \frac{x}{2}} \cdot \frac{1}{\cos ^{2} \frac{x}{2}}=$

$=\frac{-\sin ^{2} x-2 \cos ^{2} x}{\sin ^{3} x}+\frac{2 \sin ^{2} x}{\sin ^{3} x}-\frac{6}{2 \sin \frac{x}{2} \cdot \cos \frac{x}{2}}=$

$=\frac{-\sin ^{2} x-2 \cos ^{2} x+2 \sin ^{2} x}{\sin ^{3} x}-\frac{6}{\sin x}=\frac{-2 \cos ^{2} x-5 \sin ^{2} x}{\sin ^{3} x}=-\frac{2+3 \sin ^{2} x}{\sin ^{3} x}$

## Problem Kuznetsov Differentiation 15-22