Find two real numbers $\alpha$ and $\beta$ such that:
$$\forall n \in \mathbf{N}^*, \int_0^{\pi} (\alpha t^2 + \beta t) \cos(nt) \mathrm{d}t = \frac{1}{n^2}$$
then verify that if $t \in ]0, \pi]$, then:
$$\forall n \in \mathbf{N}^*, \sum_{k=1}^n \cos(kt) = \frac{\sin\left(\frac{(2n+1)t}{2}\right)}{2\sin\left(\frac{t}{2}\right)} - \frac{1}{2}$$