## Task 29/67

The equation $\sin (x+y) \sin (x-y)-\cos (x+y) \cos (x-y)=0.5$ is to be solved.
According to the addition theorem, we have

$$
\begin{aligned}
& \sin (x+y)=\sin x \cos y+\cos x \sin y \\
& \sin (x-y)=\sin x \cos y-\cos x \sin y \\
& \cos (x+y)=\cos x \cos y-\sin x \sin y \\
& \cos (x-y)=\cos x \cos y+\sin x \sin y
\end{aligned}
$$

Substituting these into the equation and multiplying out the products, we get

$$
\sin ^{2} x \cos ^{2} y-\cos ^{2} x \sin ^{2} y-\cos ^{2} x \cos ^{2} y+\sin ^{2} x \sin ^{2} y=0.5
$$

By factoring, we obtain

$$
\left(\sin ^{2} x-\cos ^{2} x\right)\left(\cos ^{2} y+\sin ^{2} y\right)=0.5
$$

Using $\sin ^{2} x+\cos ^{2} x=1$, the equation simplifies to $2 \sin ^{2} x=1.5$. It becomes clear that the equation is satisfied for any $y$. The further solution yields

$$
\sin ^{2} x=\frac{3}{4} \rightarrow|\sin x|=\frac{1}{2} \sqrt{3} \rightarrow x_{1 ; 2}=\frac{\pi}{3} \pm k \pi ; x_{3 ; 4}=\frac{2 \pi}{3} \pm k \pi
$$

$(k=1 ; 2 ; 3 ; \ldots)$ Thus, the equation holds for these four value series and for any $y$-value.