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Show that for all integers $m$ and $n,$ with $m \neq \pm n$ $\int_{-x}^{\pi} \sin m \theta \sin n \theta d \theta=0.$...

Question

Show that for all integers $m$ and $n,$ with $m \neq \pm n$ $\int_{-x}^{\pi} \sin m \theta \sin n \theta d \theta=0.$

Show that for all integers $m$ and $n,$ with $m \neq \pm n$ $\int_{-x}^{\pi} \sin m \theta \sin n \theta d \theta=0.$



Answers

Prove the formula, where $ m $ and $ n $ are positive integers.

$ \displaystyle \int_{-\pi}^{\pi} \sin mx \cos nx dx = 0 $


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