# How to prove two trigonometric identities

I want to show that $${\sin}^2 \alpha + 4{\sin}^4\frac{\alpha}{2} = 4{\sin}^2 \frac{\alpha}{2}$$ and $${\sin}^2 \alpha + 4{\cos}^4\frac{\alpha}{2} = 4{\cos}^2 \frac{\alpha}{2}$$

They should be true, as Wolfram Alpha says so. However, I want to prove them, and I have no idea how to proceed here. Any ideas?

$${\sin}^2 \alpha + 4{\sin}^4\frac{\alpha}{2}$$ $$\left(2\sin\frac {\alpha} {2}\cos \frac{\alpha}{2}\right)^2 + 4{\sin}^4\frac{\alpha}{2}$$ $$4\sin^2\frac {\alpha} {2}\cos^2 \frac{\alpha}{2} + 4{\sin}^4\frac{\alpha}{2}$$ $$4\sin^2\frac {\alpha} {2}(\cos^2 \frac{\alpha}{2} + {\sin}^2\frac{\alpha}{2})$$ $$4\sin^2\frac {\alpha} {2}\cdot 1$$ $$4\sin^2\frac {\alpha} {2}$$
Hint: $$\sin^2 \alpha=4\sin^2\frac{\alpha}2\cos^2\frac{\alpha}2$$