Question

Rewrite the expression as an equivalent expression that does not contain powers of trigonometric functions greater than 1.

Answer 1

Answer:

The expression becomes;

$\frac{3}{8}-\frac{1}{2}\cos 2x+\frac{1}{8}\cos 4x$

Explanation:

Given the trigonometric expression;

$\sin ^4x$

Simplifying and rewriting the expression;

Recall that;

$\begin{gathered} \cos 2x=1-2\sin ^2x \\ \sin ^2x=\frac{1-\cos 2x}{2} \end{gathered}$

So, the expression becomes;

$\begin{gathered} \sin ^4x=(\sin ^2x)(\sin ^2x) \\ =(\frac{1-\cos2x}{2})(\frac{1-\cos2x}{2}) \\ =(\frac{1-2\cos2x+\cos^22x}{4}) \\ =\frac{1}{4}-\frac{2}{4}\cos 2x+\frac{1}{4}\cos ^22x \end{gathered}$

Also;

$\begin{gathered} \cos 4x=2\cos ^22x-1 \\ \cos ^22x=\frac{\cos 4x+1}{2} \end{gathered}$

substituting to the above expression;

$\begin{gathered} =\frac{1}{4}-\frac{2}{4}\cos 2x+\frac{1}{4}(\frac{\cos4x+1}{2}) \\ =\frac{1}{4}-\frac{1}{2}\cos 2x+\frac{1}{8}\cos 4x+\frac{1}{8} \\ =\frac{1}{4}+\frac{1}{8}-\frac{1}{2}\cos 2x+\frac{1}{8}\cos 4x \\ =\frac{3}{8}-\frac{1}{2}\cos 2x+\frac{1}{8}\cos 4x \end{gathered}$

Therefore, the expression becomes;

$\frac{3}{8}-\frac{1}{2}\cos 2x+\frac{1}{8}\cos 4x$