Question

please show me how to use the power-reducing formulas to rewrite the expression as an equivalent expression that does not contain powers of trigonometric functions greater than 1 if you can, I have the steps already but I'm struggling to understand still.

Answer 1

Power reduction formulas for squares:

$\begin{gathered} \sin ^2u=\frac{1-\cos (2u)}{2} \\ \\ \cos ^2u=\frac{1+\cos (2u)}{2} \end{gathered}$

Given expression:

$72\sin ^2x\cos ^2x$

Use the reduction formula: For the given expression u is x:

$=72\cdot\frac{1-\cos2x}{2}\cdot\frac{1+\cos2x}{2}$

Simplify:

-Multiply:

$=\frac{72\cdot(1-\cos 2x)(1+\cos 2x)}{4}$

-Divide 72 into 4:

$=18(1-\cos 2x)(1+\cos 2x)$

Then, an equivalent expression that does not contain powers of trigonometric functions greater than 1 is:

$18(1-\cos 2x)(x+\cos 2x)$