Let be $f$ real and $ f(t)= f(-t) $
How can I show that the fourier coefficents come to $$ \hat{f} (n)= \frac{1}{ \pi} \int_0^{ \pi} f(t) \cos (nt) dt $$
I know I can rewrite $f$ as $$ f(t)= \frac{a_o}{2} + \sum_{k=1}^{ \infty} (a_k \cos(kt) + b_k \sin(kt))$$
You did a rewriting, now do it again for
$$ f(-x)\,\,=\,\,\ldots $$
Done? Then take into account @mathreadler reminder (see their comment above) when you compare the two rewritings since $\ f(-t)=f(t)$. Done? Great!