8.135. $\sin ^{4} x+\cos ^{4} x=\cos ^{2} 2 x+0.25$.
## Solution.

From the condition

$$
\begin{aligned}
& \left(\sin ^{2} x\right)^{2}+\left(\cos ^{2} x\right)^{2}=\cos ^{2} 2 x+0.25 \Leftrightarrow\left(\frac{1-\cos 2 x}{2}\right)^{2}+\left(\frac{1+\cos 2 x}{2}\right)^{2}= \\
& =\cos ^{2} 2 x+0.25 \Leftrightarrow 2 \cos ^{2} 2 x-1=0 \Leftrightarrow \cos 4 x=0,4 x=\frac{\pi}{2}+\pi k \\
& x=\frac{\pi}{8}+\frac{\pi k}{4}=\frac{\pi}{8}(2 k+1), \quad k \in Z
\end{aligned}
$$

Answer: $\quad x=\frac{\pi}{8}(2 k+1), k \in Z$.