Show that $\displaystyle \frac{1-cos2x}{1+cos2x}=tan^2x$
and hence find the exact value of tan22.5
From the formulas $\displaystyle \cos 2x = \cos^2 x -\sin^2 x$ and $\displaystyle \sin^2 x + \cos^2 x =1$:
LHS=$\displaystyle \frac{1-\cos 2x}{1+\cos 2x}=\frac{1-\cos^2 x +\sin^2 x}{1+ \cos^2 x -\sin^2 x}=\frac{2\sin^2 x}{2\cos^2 x}=\tan^2 x$=RHS.
Now plug in x=22.5 and solve for $\displaystyle \tan 22.5$.