Question

Double-angle identities: Use double-angle identities to rewrite each of the following trigonometric expressions until no exponents are involved.

${\sin }^{4}x=\_\_\_\_$

Short answer

${\sin }^{4}x=\frac{1}{4}+\frac{\cos 4x+1}{8}-\frac{\cos 2x}{2}$

Step-by-step solution

Step 1. Given information.

Consider the given information,

${\sin }^{4}x=\_\_\_\_$

Step 2. Explanation.

Write the given trigonometric expression in simplified form.

$\begin{array}{l}{\sin }^{4}x={\left({\sin }^{2}x\right)}^2\\ ={\left(\frac{1-\cos 2x}{2}\right)}^2\\ =\frac{1+{\cos }^{2}2x-2\cos 2x}{4}\\ =\frac{1}{4}+\frac{\frac{\cos 4x+1}{2}}{4}-\frac{\cos 2x}{2}\\ =\frac{1}{4}+\frac{\cos 4x+1}{8}-\frac{\cos 2x}{2}\end{array}$