Prove a fourth-power trigonometric identity

Using $\cos2\theta=2\cos^2\theta-1$, the right-hand side becomes $\dfrac18(2-4\cos\theta+2\cos^2\theta)$.

Let $s=\sin(\theta/2)$, so $\cos\theta=1-2s^2$. Then $\dfrac18[2-4(1-2s^2)+2(1-2s^2)^2]=\dfrac18[2-4+8s^2+2(1-4s^2+4s^4)]=s^4=\sin^4(\theta/2)$, proving the identity.