The space is CO,2π] and the inner product is (fg)-J 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 ntdt Use a trigonometric identity to write the integrand as a sum of sines. 2T (sin mt, cos nt) =乏J L-1 dt Then integrate with respect to t (sin mt cos nt) = 2 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