2n The space is C[0,2r] and the inner product is (f.g) = f(t)g(t) dt. Show that sin mt and cos nt are orthogonal for all positive integers m and n. Begin by writing the inner product using the given functions. (sin mt, cos nt) = dt Use a trigonometric identity to write the integrand as a sum of sines. 2x 1 (sin mt, cos nt) =- dt Then integrate with respect to t. (sin mt, cos nt) = Evaluate the result at the end points of the interval. Note that m -n in the denominator means that this result does not apply to m = n. (sin mt, cos nt) = [(O - 0) (Simplify your answers.) Then simplify this result to get the inner product for all positive integers m # n. (sin mt, cos nt) =

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