Integral Question: Going from 1st to 2nd w/ Const 'A' Divided by 2

[ZedCar]

Homework Statement

Could anyone please explain to me how the first integral in the attachment goes to the second?

Also, why the constant 'A' in the second is divided by 2?

'n' is an integer number.

Thank you.

Homework Equations

The Attempt at a Solution

 

[slider142]

They first use the half-angle identity $\cos^2(\theta) = \frac{1}{2}(1 + \cos(2\theta))$, and then use the fact that the cosine function has a period of 2π.

 

[ZedCar]

They first use the half-angle identity $\cos^2(\theta) = \frac{1}{2}(1 + \cos(2\theta))$, and then use the fact that the cosine function has a period of 2π.

Thanks slider142