Answer
$1 - 8 ~sin^2~\frac{x}{2}~cos^2~\frac{x}{2} = cos~2x$
Work Step by Step
$1 - 8 ~sin^2~\frac{x}{2}~cos^2~\frac{x}{2}$
When we graph this function, we can see that it looks like the graph of $~~cos~2x$
Note that: $~sin~2a = 2~sin~a~cos~a$
Also: $~~cos~2b = 1-2~sin^2~b$
We can verify this algebraically:
$1 - 8 ~sin^2~\frac{x}{2}~cos^2~\frac{x}{2} = 1 - 2~(4~sin^2~\frac{x}{2}~cos^2~\frac{x}{2})$
$1 - 8 ~sin^2~\frac{x}{2}~cos^2~\frac{x}{2} = 1 - 2~(2~sin~\frac{x}{2}~cos~\frac{x}{2})^2$
$1 - 8 ~sin^2~\frac{x}{2}~cos^2~\frac{x}{2} = 1 - 2~(sin~x)^2$
$1 - 8 ~sin^2~\frac{x}{2}~cos^2~\frac{x}{2} = cos~2x$
