Find the Wronskian of the given pair of functions:
$$
cos^2(x),\,1+cos(2x)
$$
$$
W=
det\begin{vmatrix}
cos^2(x)&1+cos(2x)\\
-2cos(x)sin(x)&-2sin(2x)\\
\end{vmatrix}
=
det\begin{vmatrix}
cos^2(x)&1+cos(2x)\\
-sin(2x)&-2sin(2x)\\
\end{vmatrix}\\
=-2cos^2(x)sin(2x)+sin(2x)+sin(2x)cos(2x)\\
=-sin(2x)[2cos^2(x)-1-cos(2x)]\\
=-sin(2x)[cos(2x)-cos(2x)]\\
=-sin(2x)\times0\\
=0\\
\
Note:
cos(2x)=2cos^2(x)-1\\
sin(2x)=2sin(x)cos(x)\\
$$
