## Task A-3.1.

If $\cos x + \sin x = 0.3$, find $\cos^{3} x + \sin^{3} x$.
## First Solution.

Squaring the initial equality, we get

$$
\cos ^{2} x+2 \cos x \sin x+\sin ^{2} x=0.09 \quad \quad 1 \text { point }
$$

For any real $x$, $\cos ^{2} x+\sin ^{2} x=1$. 1 point

Substituting this into the above calculation, we get $1+2 \cos x \sin x=0.09$, from which

$$
\cos x \sin x=\frac{0.09-1}{2}=-0.455
$$

Using the formula for the sum of cubes, we have

$$
\begin{aligned}
\cos ^{3} x+\sin ^{3} x & =(\cos x+\sin x)\left(\cos x^{2}-\sin x \cos x+\cos ^{2} x\right) & & 2 \text { points } \\
& =0.3 \cdot(1+0.455) & & \\
& =0.4365 . & & 1 \text { point }
\end{aligned}
$$