[Solved] The problem is an applications of the trigonometric identities sin A sin B=1/2[ cos (A-B)- cos (A+B)] , sin A cos B=1/2[ sin (A-B)+ sin (A+B)] ,
Question: The problem is an applications of the trigonometric identities
\[\begin{aligned} & \sin A\sin B=\frac{1}{2}[\cos (A-B)-\cos (A+B)] \\ & \sin A\cos B=\frac{1}{2}[\sin (A-B)+\sin (A+B)] \\ & \cos A\cos B=\frac{1}{2}[\cos (A-B)+\cos (A+B)] \\ \end{aligned}\]
Suppose that \(m\) and \(n\) are positive integers with \(m \neq n\). Show that
\[\begin{aligned} &\text { (a) } \int_{0}^{2 \pi} \sin m x \sin n x d x=0 \\ &\text { (b) } \int_{0}^{2 \pi} \cos m x \sin n x d x=0 \\ &\text { (c) } \int_{0}^{2 \pi} \cos m x \cos n x d x=0 \end{aligned}\]
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