2. The expression is

$$
A=\frac{\frac{1}{2} \sin 2 x+\cos 2 x}{1-\tan x}
$$

Calculate the value of the expression $A$ if $\cos x=-\frac{4}{5}$ and $\frac{\pi}{2}<x<\pi$. The result should be exact. (10 points)

## 19th Mathematics Knowledge Competition for Students of Secondary Technical and Vocational Schools

## Selection Competition, March 21, 2019

## Solutions to Problems and Scoring

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

A correct method is considered to be any procedure that:

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

If an intermediate or final result can be recognized, guessed, read from a diagram, or calculated mentally, the contestant generally receives all the points provided. However, if the solution is guessed (and cannot be arrived at without calculation), it is scored with 0 points.

A contestant who has only partially solved the problem, with correct procedures but no clear path to the final solution, cannot receive more than half of the possible points.

The asterisk '*' in the points indicates that the point or points can be awarded for a correct procedure, even if the calculation is incorrect.

## First Year
2. We use the relationship between trigonometric functions $\sin ^{2} x+\cos ^{2} x=1$ and get $\sin ^{2} x=\frac{9}{25}$. Since $\frac{\pi}{2}<x<\pi$, we have $\sin x=\frac{3}{5}$. We note that $\tan x=\frac{\sin x}{\cos x}$, $\sin 2 x=2 \sin x \cos x$ and $\cos 2 x=$ $\cos ^{2} x-\sin ^{2} x$. The calculated value of the expression $A=-\frac{4}{35}$.

![](https://cdn.mathpix.com/cropped/2024_06_07_18400104441c5ee1e47ag-19.jpg?height=63&width=1744&top_left_y=1462&top_left_x=173)

![](https://cdn.mathpix.com/cropped/2024_06_07_18400104441c5ee1e47ag-19.jpg?height=57&width=1742&top_left_y=1516&top_left_x=174)

![](https://cdn.mathpix.com/cropped/2024_06_07_18400104441c5ee1e47ag-19.jpg?height=57&width=1739&top_left_y=1568&top_left_x=176)

![](https://cdn.mathpix.com/cropped/2024_06_07_18400104441c5ee1e47ag-19.jpg?height=54&width=1731&top_left_y=1618&top_left_x=174)

Noting or considering $\cos 2 x=\cos ^{2} x-\sin ^{2} x=\frac{7}{25} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$

![](https://cdn.mathpix.com/cropped/2024_06_07_18400104441c5ee1e47ag-19.jpg?height=63&width=1731&top_left_y=1713&top_left_x=174)

![](https://cdn.mathpix.com/cropped/2024_06_07_18400104441c5ee1e47ag-19.jpg?height=63&width=1736&top_left_y=1762&top_left_x=177)