Consider the following integral,
I tried taking one $\cos x$ out and then integrating by parts. I also tried integrating by parts using $\cos(nx)$ as the other function but didn't get any relation between $I(n)$ and $I(n-1)$.
In both the ways, $\cos(nx)$ was creating a problem as it wasn't getting converted to $\cos[(n-1)x]$ which was required for getting a term of $I(n-1)$.
Any hint/partial solution would be appreciated.