## Task B-3.3.

If $f(x)=4 \sin ^{2} \frac{3 x}{2}-4 \cos ^{2} \frac{3 x}{2}$, determine $f\left(\frac{2020 \pi}{9}+2021 k \pi\right)$ depending on the integer $k$.
## Solution.

The function $f$ can be written in the following form:

$$
f(x)=4 \sin ^{2} \frac{3 x}{2}-4 \cos ^{2} \frac{3 x}{2}=-4\left(\cos ^{2} \frac{3 x}{2}-\sin ^{2} \frac{3 x}{2}\right)=-4 \cos 3 x
$$

Therefore,

$$
f\left(\frac{2020 \pi}{9}+2021 k \pi\right)=-4 \cos \left(3\left(\frac{2020 \pi}{9}+2021 k \pi\right)\right)=-4 \cos \left(\frac{2020 \pi}{3}+6063 k \pi\right)
$$

We have two cases.

$1^{\circ}$ if $k$ is even, the above expression is equal to

$$
-4 \cos \left(\frac{2020 \pi}{3}\right)=-4 \cos \left(\frac{4 \pi}{3}\right)=-4\left(-\frac{1}{2}\right)=2
$$

$2^{\circ}$ if $k$ is odd, the above expression is equal to

$$
-4 \cos \left(\frac{2020 \pi}{3}+\pi\right)=-4\left(-\cos \left(\frac{4 \pi}{3}\right)\right)=4 \cos \left(\frac{4 \pi}{3}\right)=4\left(-\frac{1}{2}\right)=-2
$$