B4. Given the function $f(x)=\frac{\sin x-\sin ^{3} x}{\cos ^{3} x}, \cos x \neq 0$.

a) Simplify the expression $\frac{\sin x-\sin ^{3} x}{\cos ^{3} x}, \cos x \neq 0$.

b) Calculate $f\left(\frac{4 \pi}{3}\right)$.

c) Calculate the zeros of the function $f$.

## Solutions and Scoring

A contestant who arrives at the solution by any correct method (even if not anticipated by the scoring) receives all possible points.

A correct method is one that

- sensibly takes into account the wording of the problem,
- leads to the solution of the problem,
- is mathematically correct and complete.

A contestant who has only partially solved the problem, from otherwise correct solving procedures but does not show the path to the final solution of the problem, cannot receive more than half of the possible points.

## First Year

I. PART

| Problem | 1 | 2 | 3 | 4 | 5 | 6 |
| :--- | :---: | :---: | :---: | :---: | :---: | :---: |
| Answer | $\mathrm{D}$ | $\mathrm{C}$ | $\mathrm{B}$ | $\mathrm{D}$ | $\mathrm{B}$ | $\mathrm{E}$ |
B4. We simplify the functional expression by first factoring out the common factor $f(x)=\frac{\sin x\left(1-\sin ^{2} x\right)}{\cos ^{3} x}$. We note that $1-\sin ^{2} x=\cos ^{2} x$. We rewrite $f(x)=\tan x$. We calculate the value of the function: $f\left(\frac{4 \pi}{3}\right)=\tan \frac{4 \pi}{3}=\sqrt{3}$. We find the zeros from $\tan x=0$ or $\sin x=0$. We get the solution $x_{k}=k \cdot \pi, k \in \mathbb{Z}$.

![](https://cdn.mathpix.com/cropped/2024_06_07_bce9c6e2a6f4f88bb0deg-16.jpg?height=75&width=1641&top_left_y=1573&top_left_x=276)

![](https://cdn.mathpix.com/cropped/2024_06_07_bce9c6e2a6f4f88bb0deg-16.jpg?height=58&width=1639&top_left_y=1636&top_left_x=274)

![](https://cdn.mathpix.com/cropped/2024_06_07_bce9c6e2a6f4f88bb0deg-16.jpg?height=52&width=1639&top_left_y=1690&top_left_x=274)

The calculated value $f\left(\frac{4 \pi}{3}\right)=\tan \frac{4 \pi}{3}=\sqrt{3} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . \ldots$ point

![](https://cdn.mathpix.com/cropped/2024_06_07_bce9c6e2a6f4f88bb0deg-16.jpg?height=54&width=1637&top_left_y=1789&top_left_x=278)

![](https://cdn.mathpix.com/cropped/2024_06_07_bce9c6e2a6f4f88bb0deg-16.jpg?height=51&width=1637&top_left_y=1842&top_left_x=278)