Hi Liz,

The expression I remember for cos(2x) is

\[\cos(2x) = \cos^2(x) - \sin^2(x).\]

Use the fact that $\sin^2(x) + \cos^2(x) = 1$ to write the expression for $\cos(2x)$ in terms of $\sin(x)$ alone. Substitute into your equation $\cos(2x) = \sin(x).$ This gives a quadratic equation in $\sin(x).$ solve the quadratic and then solve for $x.$

To check that you have all the solutions I would graph $y = \cos(2x)$ and $y = \sin(x)$ to see where they intersect.

Penny