Task B-3.3. What is $f\left(\frac{\pi}{32}\right)$ if $f(x)=\frac{\sin 12 x}{\sin 4 x}+\frac{\cos 12 x}{\cos 4 x}$?
## Solution.

Simplify the expression given for the function $f$.

$$
\begin{aligned}
f(x) & =\frac{\sin 12 x}{\sin 4 x}+\frac{\cos 12 x}{\cos 4 x}=\frac{\sin 12 x \cdot \cos 4 x+\cos 12 x \cdot \sin 4 x}{\sin 4 x \cos 4 x} \\
& =\frac{\sin 16 x}{\frac{1}{2} \sin 8 x} \\
& =\frac{2 \cdot 2 \sin 8 x \cos 8 x}{\sin 8 x}=4 \cos 8 x
\end{aligned}
$$

Then,

$$
f\left(\frac{\pi}{32}\right)=4 \cos \left(8 \cdot \frac{\pi}{32}\right)=4 \cos \frac{\pi}{4}=4 \cdot \frac{\sqrt{2}}{2}=2 \sqrt{2}
$$