Find the Wrouskian of the given parts of function
$$
\cos ^{2}(x) \quad 1+\cos (2 x)
$$
$$
\begin{aligned}
w&=\operatorname{det}\left|\begin{array}{cc}{\cos ^{2}(x)} & {1+\cos (2 x)} \\ {-2 \cos (x) \sin (x)} & {-2 \sin (2 x)}\end{array}\right|\\
&=-2 \sin (2 x) \cos ^{2}(x)+[2 \cos (x) \sin (x)][1+\cos (2 x)]\\
&=-2 \sin (2 x) \cos ^{2}(x)+2 \cos (x) \sin (x)+2 \cos (x) \sin (x) \cos (2 x)\\
&=-2 \sin (2 x) \cos ^{2}(x)+\sin (2 x)+\sin (2 x) \cos (2 x)\\
&=\sin (2 x)\left(-2 \cos ^{2}(x)+1+\cos (2 x)\right)\\
&=\sin (2 x)\left(1-2 \cos ^{2}(x)+2 \cos ^{2}(x)-1\right)\\
&=0
\end{aligned}
$$
