8.329. $2 \cos ^{2} \frac{x}{2}(1-\sin x)+\cos ^{2} x=0$.

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8.329. $2 \cos ^{2} \frac{x}{2}(1-\sin x)+\cos ^{2} x=0$.
## Solution.

The sum of two non-negative numbers is zero if and only if each of them is zero. Thus, we have:

$$
\begin{aligned}
& \left\{\begin{array} { l } 
{ 2 \operatorname { cos } \frac { 2 x } { 2 } ( 1 - \operatorname { sin } x ) = 0 , } \\
{ \operatorname { cos } ^ { 2 } x = 0 , }
\end{array} \Leftrightarrow \left\{\begin{array} { l } 
{ ( 1 + \operatorname { cos } x ) ( 1 - \operatorname { sin } x ) = 0 , } \\
{ \operatorname { cos } x = 0 , }
\end{array} \Leftrightarrow \left\{\begin{array}{l}
\sin x=1, \\
\cos x=0,
\end{array} \Leftrightarrow\right.\right.\right. \\
& x=\frac{\pi}{2}+2 \pi k=\frac{\pi}{2}(4 k+1), k \in Z \\
& \text { Answer: } x=\frac{\pi}{2}(4 k+1), k \in Z
\end{aligned}
$$