How do you graph $\displaystyle cos^2 x$?
[we use a TI-84+ in school so that's easier...]
There are several substitution formulas:
$\displaystyle cos^2 x + sin^2 x = 1$
$\displaystyle 2cos^2 x - 1 = cos2 x$
$\displaystyle 1 - 2sin^2 x = cos2x$
$\displaystyle cos^2 x - sin^2 x = cos2x$
If you use the second one you get
$\displaystyle 2cos^2 x - 1 = cos2 x$
$\displaystyle 2cos^2 x = 1 + cos2x$
$\displaystyle cos^2 x = 1/2 + 1/2 cos2x$
so the problem changed to how do you draw the graph of $\displaystyle 1/2 + 1/2cos2x$