QuestionUse half angle formula $$(\sin (2 x))^{4}=$$

Studdy Solution
Use the identity $\sin^2(x) = \frac{1 - \cos(2x)}{2}$ to express $(\sin(x)\cos(x))^2$:
$$(\sin(x)\cos(x))^2 = \sin^2(x)\cos^2(x)$$
$$= \left(\frac{1 - \cos(2x)}{2}\right)\left(\frac{1 + \cos(2x)}{2}\right)$$
$$= \frac{1 - \cos^2(2x)}{4}$$
$$= \frac{\sin^2(2x)}{4}$$
Since $\sin^2(2x) = 1 - \cos^2(2x)$, substitute back:
$$(\sin(x)\cos(x))^4 = \left(\frac{\sin^2(2x)}{4}\right)^2$$
$$= \frac{(\sin^2(2x))^2}{16}$$
$$= \frac{(1 - \cos^2(2x))^2}{16}$$
The simplified expression for $(\sin(2x))^4$ using the half-angle formula is:
$$\boxed{\frac{(1 - \cos^2(2x))^2}{16}}$$