7. If the equation $\cot (\arcsin x)=\sqrt{a^{2}-x^{2}}$ has real solutions for $x$, then the range of real values for $a$ is $\qquad$ .
7. $(-\infty,-1] \cup[1,+\infty)$.

Let $\alpha=\arcsin x$, then $x=\sin \alpha, -1 \leqslant x \leqslant 1$. According to the problem, $\cot \alpha \geqslant 0$, so $\cot \alpha = \sqrt{\csc^2 \alpha - 1} = \sqrt{\frac{1-x^2}{x^2}}$, thus $\sqrt{\frac{1-x^2}{x^2}} = \sqrt{a^2 - x^2} \Rightarrow x^4 - (a^2 + 1)x^2 + 1 = 0$. According to this equation having real solutions, we get $\Delta = (a^2 + 1)^2 - 4 \geqslant 0, a^2 + 1 \geqslant 2 \Rightarrow a^2 \geqslant 1$. Therefore, $\alpha \in (-\infty, -1] \cup [1, +\infty)$.