math

Use integration by parts to verify
|(from -pi to +pi)cosmxcosnxdx = pi when m=n

and

|(from -pi to +pi)cosmxcosnxdx = 0 when m and n are not equal.

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  1. Assume that for all cases that both m and n are integers.

    For m = n:
    |(from -pi to +pi) cos^2(mx) dx =
    = |(from -pi to +pi) (1/2)(1 + cos(2mx)) dx
    = (1/2)(pi + pi) + (1/4){sin(2mpi) - sin(-2mpi)}
    = pi

    For m not equal n:
    |(from -pi to +pi) cos(mx) cos(nx) dx =
    = |(from -pi to +pi) (1/2)*{cos(m+n)x + cos(m-n)x} dx
    = (1/2(m+n))*(sin(m+n)pi - sin(-(m+n)pi)) + (1/2(m-n))*(sin(m-n)pi - sin(-(m-n)pi))
    = 0

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