10. -/1 points details my notes oscat1 9.3.145. use the power - reduction formulas to rewrite the…

Explanation:

Step1: Recall power - reduction formulas

The power - reduction formulas are $\sin^{2}\theta=\frac{1 - \cos(2\theta)}{2}$ and $\cos^{2}\theta=\frac{1+\cos(2\theta)}{2}$.

Step2: Rewrite $3\sin^{2}(x)\cos^{2}(x)$

Substitute the power - reduction formulas into the expression: [ \begin{align*} 3\sin^{2}(x)\cos^{2}(x)&=3\times\frac{1 - \cos(2x)}{2}\times\frac{1+\cos(2x)}{2}\ &=\frac{3}{4}(1-\cos^{2}(2x)) \end{align*} ] Then, use the power - reduction formula for $\cos^{2}(2x)=\frac{1 + \cos(4x)}{2}$: [ \begin{align*} \frac{3}{4}(1-\cos^{2}(2x))&=\frac{3}{4}\left(1-\frac{1 + \cos(4x)}{2}\right)\ &=\frac{3}{4}\times\frac{2-(1 + \cos(4x))}{2}\ &=\frac{3}{8}(1-\cos(4x)) \end{align*} ]

Step3: Rewrite $\sin^{2}(5x)$

Using the power - reduction formula $\sin^{2}\theta=\frac{1 - \cos(2\theta)}{2}$ with $\theta = 5x$, we get $\sin^{2}(5x)=\frac{1-\cos(10x)}{2}$.

Answer:

For $3\sin^{2}(x)\cos^{2}(x)$: $\frac{3}{8}(1 - \cos(4x))$ For $\sin^{2}(5x)$: $\frac{1-\cos(10x)}{2}$