Solution:

$$\begin{aligned} \mathrm{LHS} &=\frac{\sin 4 \theta+\sin 2 \theta}{1+\cos 4 \theta+\cos 2 \theta} \ &=\frac{2 . \sin 2 \theta \cdot \cos 2 \theta+\sin 2 \theta}{2 \cos ^{2} 2 \theta+\cos 2 \theta} \ &=\frac{\sin 2 \theta(2 \cos 2 \theta+1)}{\cos 2 \theta(2 \cos 2 \theta+1)} \ &=\tan 2 \theta \end{aligned}$$