1. For which values of $a$ does the equation

$$
1+\sin ^{2} a x=\cos x
$$

have a unique solution?
Solution. From the given equation and from $\sin ^{2} a x \geq 0, \cos x \leq 1$, we get $\sin ^{2} a x=0$ and $\cos x=1$, from which we obtain

$$
a x=k \pi, \quad x=2 k \pi
$$

where $k$ and $m$ are integers.

If $a$ is an irrational number, then from $a \cdot 2 m=k$ we get $m=k=r$, so the given trigonometric equation will have the unique solution $x=0$.

If $a$ is a rational number, from $a x=k \pi$ and $x=2 m \pi$, it is clear that the equation will have infinitely many solutions.

Therefore, the equation will have a unique solution if $a$ is an irrational number.