Problem

The key insight is that if an expression involving $\theta$ equals zero for all values of $\theta$, then each independent component must vanish separately. Rewrite the equation by expressing everything in terms of $\cos 2\theta$ and $\sin^2 \theta$ (or equivalently, collect coefficients of linearly independent trigonometric functions). Since the equation must hold for every $\theta$, the coefficient of each independent term—and the constant term—must all equal zero. This gives you a system of linear equations in $x$, $y$, and $z$ to solve.

The key insight is that if an expression involving $\theta$ equals zero for all values of $\theta$, then each independent component must vanish separately. Rewrite the equation by expressing everything in terms of $\cos 2\theta$ and $\sin^2 \theta$ (or equivalently, collect coefficients of linearly independent trigonometric functions). Since the equation must hold for every $\theta$, the coefficient of each independent term—and the constant term—must all equal zero. This gives you a system of linear equations in $x$, $y$, and $z$ to solve.