Simplifying $\sin(4x)\cos(4x)$
Solution 1:
$\sin4x\cos4x$ using identiity that $\sin(2\theta)=2\sin\theta\cos\theta$,we get
$\sin4x\cos4x=\frac{\sin(8x)}{2}$
Solution 2:
The double angle formula is $ \ 2 \sin\theta\cos\theta = \sin(2\theta) \iff \sin\theta\cos\theta = \frac{1}{2} \sin(2\theta)$.
By applying this formula with $\theta = 4x$, we obtain
$$\sin4x\cos4x=\frac{1}{2} \sin(8x).$$