How do I approximate the inner product of trigonometric functions with a small perturbation?

Question

I learned that $\int_0^1 \cos(n \pi t)\cos(m \pi t)~\mathrm{d}t = 1/2$ for any positive $m$ and $n$, becuase of the orthogonality of trigonometric functions. If one of the basis function $\cos(m\pi t)$ has a small perturbation (i.e. $\cos(m \pi (t+\epsilon f(t)))$), I was wondering if I could approximate such a integral by using something like: $$ \int_0^1 \cos(n \pi t)\cos(m \pi (t+\epsilon f(t)))~\mathrm{d}t\approx\frac{1}{2}+\epsilon g_1(m,n)+\cdots $$ where $n\geq 1$ and $m\geq 1$ are positive integers, and $|f(t)|<1$ for any $0\leq t\leq 1$ with $\epsilon\ll 1$. If so, how could we obtain the expression of unknown function $g_1(m,n)$?