How may I simplify the following trigonomic expression;

Question

How may I simplfy for the general equation showing the values of theta that satisfy the following;

$ \{\theta \in \mathbb R \mid 2\sin^4(\theta) = \cos^2(\theta) \}$

Answer 1

$$2s^2=1-s\iff0=2s^2+s-1=(2s-1)(s+1)$$

As $s=\sin^2\theta\ge0$ for real $\sin\theta$

$2s-1=0\implies\cos2(\theta)=0$

$2\theta$ must be odd multiple of $\dfrac\pi2$

Answer 2

Using $\cos^2 θ =1-\sin^2 θ$ we get: $2\sin^4(θ) = \cos^2(θ) \rightarrow 2\sin^4(θ)+\sin^2(θ)-1=0 \rightarrow \sin^2(θ)=0.5$

Thus, $\large{θ=\frac{\pi}{4}+n\frac{\pi}{2}}$