Question

Prove the following identities

Answer 1

\[\cos2 \theta + \cos 4 \theta + \cos 6 \theta + \cos 12 \theta = 4 \cos 3 \theta \cos 4 \theta \cos 5 \theta\]

Answer 2

We would have to use this formula repeatedly:

\[\large{\cos x + \cos y = 2\cos(\cfrac{x+y}{2}) \cos(\cfrac{x-y}{2})}\]

Answer 3

ok..

Answer 4

Use this formula for:

\[\large{\cos{2\theta} + \cos{6\theta},~~\cos{4\theta} + \cos{12\theta}}\]

Answer 5

I am solving for the first sum and you solve for the second one

Answer 6

I am pretty sure there is a shortcut formula for this type of sum though :)

Answer 7

But lets move on...

Answer 8

\[\large{\cos{6\theta} + \cos{2\theta}}\]

\[\large{=2\cos(\cfrac{6\theta+2\theta}{2}) \cos(\cfrac{6\theta - 2\theta}{2})}\]

Answer 9

\[\large{= 2\cos{4\theta}\cos{2\theta}}\]