Prove: $\sin 2 \theta \;\ge\; \frac{1}{2}\left(-1-3 \cos^2 \theta\right)$ Please help me prove the following inequality: 

$$\sin 2 \theta \;\ge\; \frac{1}{2}\left(-1-3 \cos^2 \theta\right)$$

I have been working on it for an hour in vain. I have derived $2\sin \theta \cos \theta$ from the left side and $\frac{1}{2}\left(-4+3 \sin^2 \theta\right)$ from the right side, but I don't know where to go from there.

We need to establish $$\sin2\theta\ge-\dfrac{1+3\cos^2\theta}2=-\dfrac{2+3(1+\cos2\theta)}4$$
$$\iff4\sin2\theta+3\cos2\theta\ge-5$$
Now
$4\sin2\theta+3\cos2\theta=5\sin\left(2\theta+\arcsin\dfrac45\right)$
Finally for real $\theta,$ $$-1\le\sin\left(2\theta+\arcsin\dfrac45\right)\le1$$
