A geometry problem by Jonathan Ng

We use the identity $tan^2x+1 = sec^2x$ , or, $tan^2x = sec^2x-1 = \frac{1}{cos^2x}-1$ .

It is a (not-so-well-known) fact that $cos 36^\circ = \frac{\sqrt{5}+1}{4}$ . It is another fact that $cos 72^\circ = \frac{\sqrt{5}-1}{4}$

So, our equation becomes

$tan^2 36^\circ+tan^2 72^\circ = \frac{1}{cos^236^\circ}-1 +\frac{1}{cos^272^\circ}-1 = (\frac{4}{\sqrt{5}-1})^2+(\frac{4}{\sqrt{5}+1})^2 -2$ .

Rationalizing the denominator and expanding, we get that the expression is equal to $\boxed{10}$ .