TITLE: $(\tan^2(18^\circ))(\tan^2(54^\circ))$ is a rational number
QUESTION [5 upvotes]: Assuming $$\cos(36^\circ)=\frac{1}{4}+\frac{1}{4}\sqrt{5}$$ How to prove that         $$\tan^2(18^\circ)\tan^2(54^\circ)$$ is a rational number? Thanks!

REPLY [3 votes]: $$\tan18^\circ\tan54^\circ=\frac{2\sin18^\circ\sin54^\circ}{2\cos18^\circ\cos54^\circ}$$
$$=\frac{\cos36^\circ-\cos72^\circ}{\cos36^\circ+\cos72^\circ}$$ (applying $\cos(A\pm B)$ formulae)
$$=\frac{\frac{\sqrt5+1}4-\frac{\sqrt5+1}4}{\frac{\sqrt5+1}4+\frac{\sqrt5+1}4}$$ as $$\cos 72^\circ=2\cos^236^\circ-1=\frac{\sqrt5-1}4$$
$$\implies \tan18^\circ\tan54^\circ=\sqrt5$$