Chapter 3 - Second Order Linear Equations - 3.2 Solutions of Linear Homogenous Equations; the Wronskian - Problems - Page 155: 6

$0$

$W(cos^{2}\theta,1+cos{2\theta}) = |\begin{matrix} cos^{2}\theta & 1+cos{2\theta}\\ -2cos{\theta}sin{\theta} & -2sin{2\theta}\\ \end{matrix}| = -2sin{2\theta}cos^{2}{\theta}+sin{2\theta}(1+cos{2\theta})= -sin{2\theta}(cos{2\theta}+1)+sin{2\theta}(1+cos{2\theta})=0$