2. The solution to the equation $\sin ^{4} x-\cos ^{4} x=1$ is
2. $x=k \pi+\frac{\pi}{2}(k \in \mathbf{Z})$.

From the equation, we get
$$
\begin{array}{l}
\sin ^{4} x=1+\cos ^{4} x \geqslant 1 \\
\Rightarrow \cos x=0, \sin x= \pm 1 \\
\Rightarrow x=k \pi+\frac{\pi}{2}(k \in \mathbf{Z}) .
\end{array}
$$