QUESTION: I am trying to show that
$$a_n=-\frac {1}{\pi } \int_{-\pi }^\pi f \left( t+ \frac {\pi }{n} \right) \cos(nt) \,{\rm d}t$$
will transform to this expression with a cosine shift, but I am uncertain how to do this here to obtain:
$$a_n=\frac 1 2 \frac {1}{\pi }\int_{-\pi }^\pi \left[f(x) -f \left( x+\frac {\pi }{n} \right) \right] \cos(nx) \, {\rm d}x $$
The original definition of $a_n$ is
$$a_n = \frac{1}{\pi}\int_{-\pi}^\pi f(x)\cos(nx)\,{\rm d}x$$
and the equation you have derived reads
$$a_n = -\frac{1}{\pi}\int_{-\pi}^\pi f\left(x+\frac{\pi}{n}\right)\cos(nx)\,{\rm d}x$$
Adding these two equations gives us the desired result
$$2a_n = \frac{1}{\pi}\int_{-\pi}^\pi \left[f(x) - f\left(x+\frac{\pi}{n}\right)\right]\cos(nx)\,{\rm d}x$$
