2. The solution set of the equation about the real number $x$
$$
2^{1-2 \sin ^{2} x}=2+\log _{2}\left(1-\sin ^{2} x\right)
$$

is . $\qquad$
2. $\left\{x \mid x=k \pi\right.$ or $\left.\frac{k \pi}{2}+\frac{\pi}{4}, k \in \mathbf{Z}\right\}$.

Let $t=\cos 2 x$. Then $t \in(-1,1]$.
The original equation transforms into an equation about $t$:
$$
2^{t}=1+\log _{2}(1+t) \text {. }
$$

When $0<t+1>1+\log _{2}(1+t) \text {. }
$$

Therefore, the solution to the equation is only $t=0$ or 1, at this time,
$$
x=k \pi \text { or } \frac{k \pi}{2}+\frac{\pi}{4}(k \in \mathbf{Z}) \text {. }
$$