Off the top of my head, perhaps a proof along the following lines:

Using the double angle formula:

.

The sum of 1's certainly diverges. If the sum of 's converges then overall you've got divergence ..... On the other hand, if twice the sum of 's diverges, it'll diverge much more slowly than the sum of 1's and so you've got divergence again ....

Sloppy I know (and the 'other hand' claim might prove to be bogus) - over to you (or others) to dot the i's and cross the t's (and put in the missing words and supply the punctuation)