## Task Condition

Find the derivative.

$$
y=\sqrt{\tan 4}+\frac{\sin ^{2} 21 x}{21 \cos 42 x}
$$
## Solution

$$
\begin{aligned}
& y^{\prime}=\left(\sqrt{\tan 4}+\frac{\sin ^{2} 21 x}{21 \cos 42 x}\right)^{\prime}=0+\left(\frac{\sin ^{2} 21 x}{21 \cos 42 x}\right)^{\prime}= \\
& =\frac{2 \sin 21 x \cdot \cos 21 x \cdot 21 \cdot \cos 42 x-\sin ^{2} 21 x \cdot(-\sin 42) \cdot 42}{21 \cos ^{2} 42 x}= \\
& =\frac{\sin 42 x \cdot \cos 42 x+2 \sin ^{2} 21 x \cdot \sin 42}{\cos ^{2} 42 x}= \\
& =\frac{\sin 42 x \cdot\left(\cos 42 x+2 \sin ^{2} 21 x\right)}{\cos ^{2} 42 x}= \\
& =\frac{\sin 42 x \cdot\left(1-2 \sin ^{2} 21 x+2 \sin ^{2} 21 x\right)}{\cos ^{2} 42 x}= \\
& =\frac{\sin 42 x}{\cos ^{2} 42 x}=\frac{\tan ^{2} 2x}{\cos 42 x}
\end{aligned}
$$