Let y = 1 - cos x / 1 + cosx
Show that dy/dx = t+t^3 , where t = tan(x/2)
Probably easiest to get y in terms of half x right away, using
$\displaystyle \cos(2x) \equiv \cos^2 x - \sin^2 x$
i.e. $\displaystyle \displaystyle{\cos(x) \equiv \cos^2 \frac{x}{2} - \sin^2 \frac{x}{2}\displaystyle{$
Then simplify using Pythag,
$\displaystyle \cos^2 x + \sin^2 x \equiv 1$