Question

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

${\sin }^{2}x=\_\_\_\_\_$

Short answer

${\sin }^2\theta =\frac{1-\cos 2\theta }{2}$

Step-by-step solution

Step 1. Given information.

Consider the given expression,

${\sin }^{2}x$

Step 2. Explanation.

Write the given trigonometric expression in cosine form.

$\begin{array}{rcl}\cos 2\theta & =& \cos \theta \cos \theta -\sin \theta \sin \theta \\ & =& {\cos }^2\theta -{\sin }^2\theta \\ & =& 1-{\sin }^2\theta -{\sin }^2\theta \\ \cos 2\theta & =& 1-2{\sin }^2\theta \\ 2{\sin }^2\theta & =& 1-\cos 2\theta \\ {\sin }^2\theta & =& \frac{1-\cos 2\theta }{2}\end{array}$