Trigonometric Integral problem

[lanmonster]

Hello, I am trying to find

∫cos(7Θ)cos(5Θ)dΘ

but I am unsure how to begin. My professor has given the solution and he says to use a trig identity to rewrite as

(1/2)∫cos(2Θ)+cos(12Θ)dΘ

and from there solve as

(1/2)((1/2)sin(2Θ) + (1/12)sin(12Θ)) + C

I understand how he goes from the rewriting to the solution, but I don't get how he got to the rewriting. What trig identity is used?
Sorry if the formatting is weird; this is my first post.

 

[Deleted member 4993]

Hello, I am trying to find

∫cos(7Θ)cos(5Θ)dΘ

but I am unsure how to begin. My professor has given the solution and he says to use a trig identity to rewrite as

(1/2)∫cos(2Θ)+cos(12Θ)dΘ

and from there solve as

(1/2)((1/2)sin(2Θ) + (1/12)sin(12Θ)) + C

I understand how he goes from the rewriting to the solution, but I don't get how he got to the rewriting. What trig identity is used?
Sorry if the formatting is weird; this is my first post.

Hint:

Cos(A) * Cos(B) = 1/2 * [ Cos(A+B) + Cos(A-B)]

 

[lanmonster]

Thank you!