Calculate expression: $\cos\alpha-\cos2\alpha$ I attempt calculate this expression:
$$\cos\dfrac{\pi}{5}-\cos\dfrac{2\pi}{5}$$
Please, help me, somebody. Thanks.

$\cos\dfrac{\pi}{5}=\cos36^\circ$ and the next is $\cos\dfrac{2\pi}{5}=\cos72^\circ$. Now so by converting into $\sin$ we have to do
$$\sin54^\circ-\sin18^\circ,$$
which can be done by following. 
Let $\alpha=18^\circ$, therefore $5\alpha=90^\circ$. So $\sin2\alpha=\sin(90^\circ-3\alpha)$, so $\sin2\alpha=\cos3\alpha$. Expanding both sides, and converting to $\sin$ we have:
$$4\sin^{2}\alpha+3\sin\alpha-1=0.$$ Hence, $\sin\alpha=\dfrac{\sqrt{5}-1}{4}$. Similarly find 
$$\sin3\alpha=\dfrac{\sqrt{5}+1}{4}.$$
So answer is 
$$\dfrac{\sqrt{5}+1}{4}-\dfrac{\sqrt{5}-1}{4}=\dfrac{1}{2}.$$
