I want to find the value of $$\sum_{k=0}^\infty \cos\left[\left(k+\frac{1}{2}\right)\pi x\right]\cos\left[\left(k+\frac{1}{2}\right)\pi t\right].$$
I think it should relate to delta function because we can use a trig identity to get: $$\sum_{k=0}^\infty \frac{1}{2}\left(\cos\left[\left(k+\frac{1}{2}\right)\pi(t+x)\right]-\cos\left[\left(k+\frac{1}{2}\right)\pi(t-x)\right]\right)$$, but how to deal with the $\frac{1}{2}\pi (t+x)$?