Is f'(x) = -2sinx + 2cos2x the derivative of f(x) = 2cosx + sin2x ? I have no way to check if I am correct.
f'(x)= 2cos(2x) - 2sin(x) = 0
$\displaystyle 2cos(2x) = 2sin(x)$
We know that 2 will cancel and $\displaystyle cos(2x) = 1-2sin^2(x)$. Substitute and move sin(x) back to the other side.
$\displaystyle 1-2sin^2(x)-sin(x)=0 \rightarrow 2sin^2(x)+sin(x) - 1 = 0$
Let u = sin(x) and solve the quadratic for u then do arcsin(u).