$\text{Find the Wronskian of the given pair of functions}$
$\text{$cos^2(x)$, $1+cos(2x)$}$
$$\quad
W= \begin{vmatrix}
cos^2(x) & 1+cos(2x) \\
-2cos(x)sin(x) & -2sin2x \\
\end{vmatrix}
$$
$$
= \begin{vmatrix}
cos^2(x) & 1+cos(2x) \\
-sin2x & -2sin2x \\
\end{vmatrix}
$$
$$\qquad\qquad\qquad\quad
=-2cos^2(x)sin(2x)-(-sin(2x))(1+cos(2x))
$$
$$\qquad\quad
=(-sin2x)(2cos^2(x)-1-cos(2x))
$$
$$\qquad\qquad\quad
=(-sin2x)(2cos^2(x)-1-2cos^2(x)+1)
$$
$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
=(-sin2x)ยท0
\\$
$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
=0
$
