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Question 6.8.3: Let C[0, 2π] have the inner product (6), and let m and n be ...

Let C[0, 2π] have the inner product (6), and let m and n be unequal positive integers. Show that \cos m t \text { and } \cos n t are orthogonal.

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\text { Use a trigonometric identity. When } m \neq n \text {, }

\langle\cos m t, \cos n t\rangle=\int_{0}^{2 \pi} \cos m t \cos n t d t.

=\frac{1}{2} \int_{0}^{2 \pi}[\cos (m t+n t)+\cos (m t-n t)] d t.

=\left.\frac{1}{2}\left[\frac{\sin (m t+n t)}{m+n}+\frac{\sin (m t-n t)}{m-n}\right]\right|_{0} ^{2 \pi}=0.

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