Series with cosine explanation

Question

Would someone please explain how I would solve this problem without having to use Wolfram Alpha? This problem was originally posted on Google Plus.

Wolfram Alpha: http://wolfr.am/Z42ivS

Answer 1

$\sum\dfrac{1}{\cos a_n\cos a_{n+1}}=\sum \dfrac{\tan a_{n+1}-\tan{a_n}}{\sin (a_{n+1}-a_n)}$

(As,$\sin(x-y)=\sin x \cos y-\sin y \cos x$)

You can finish it from here I think.