Show that $4\cos\frac\pi5\cdot\sin\frac\pi{10}=1$ We have to prove this identity : $$4\cos\frac\pi5\cdot\sin\frac\pi{10}=1$$
The book's hint is we somehow find out that $\displaystyle1+2\left(\cos\frac\pi5-\cos\frac{2\pi}5\right)$ equals something (from the geometry of a regular pentagon) and the result follows. But I can neither see why the book's hint is true nor how it could yield a result for the original problem. So could you help me proving it either by the book's hint or another way?

HINT:
Notice $$LHS=2\cos \frac{\pi}{5}\sin\frac{\pi}{10}$$
$$=2\cos 36^\circ \sin18^\circ$$  Setting the values of $\cos 36^\circ$ & $\sin 18^\circ$ 
$$=2\left(\frac{\sqrt 5+1}{4}\right)\left(\frac{\sqrt 5-1}{4}\right)$$
$$=\frac{(\sqrt 5+1)(\sqrt 5-1)}{8}$$
$$=\frac{(\sqrt 5)^2-1^2}{8}$$ $$=\frac{5-1}{8}=\frac{1}{2}$$
