How to solve this integral with two pole point and imaginary unit i?

Question

There is a integral, when $m+n$ is odd $$ \int^{\infty}_{-\infty} {\rm d}t \ \frac{ \left\{1-(-1)^{n}\exp[-it]\right\} \left\{1-(-1)^{m}\exp[it]\right\} }{(t^{2}-n^{2}\pi^{2})(t^{2}-m^{2}\pi^{2})}t = \frac{4i}{(n^{2}-m^{2})\pi}$$ and when $n+m$ is even $$ \int^{\infty}_{-\infty} {\rm d}t \ \frac{ \left\{1-(-1)^{n}\exp[-it]\right\} \left\{1-(-1)^{m}\exp[it]\right\} }{(t^{2}-n^{2}\pi^{2})(t^{2}-m^{2}\pi^{2})}t = 0$$ I don't know how to prove this integral and the answer may not right. Thank you very much.