## Task 32/78

Determine all real solutions of the equation $\quad 2 \sin ^{2}(\pi x+y) \cos \left(y+\frac{\pi}{2}\right)=x^{2}+\frac{1}{x^{2}}$
Due to $x^{2}+\frac{1}{x^{2}} \geq 2$ (the sum of two positive, reciprocal numbers is never less than 2) and

$$
2 \sin ^{2}(\pi x+y) \cos \left(y+\frac{\pi}{2}\right) \leq 2
$$

it follows immediately that

$$
x^{2}+\frac{1}{x^{2}}=2 \quad ; \quad x= \pm 1 \quad ; \quad \sin ^{2}(y \pm \pi)=1 \quad ; \quad \cos \left(y+\frac{\pi}{2}\right)=1
$$

From the last of these equations, we get $y=\frac{3 \pi}{2}+2 k \pi$ with $k=0 ; \pm 1 ; \pm 2 ; \ldots$ Indeed, $\sin ^{2}\left(\frac{3 \pi}{2} \pm \pi \pm 2 k \pi\right)=1$