I'm working through a textbook on Fourier series ("The Intuitive Guide to Fourier Analysis and Spectral Estimation: with Matlab", by Langton and Levin), and I'm a bit stuck on the derivation of one equation which is the integral of the product of two sine waves of identical frequency over a full period from time t=0 to t=$T_0$, which the book gives as:
$$\int^{T_0}_0b_k\sin(k\omega_0t)\times\sin(k\omega_0t)=b_k\frac{T_0}2$$
$b_k$ here is the coefficient or amplitude for this Fourier harmonic.
My integration is very rusty, so I'm not sure how this result was obtained. I have tried working it out by hand using this relationship from a list of trignometric integrals:
$$\int \sin^2(ax) dx=\frac x2 -\frac1{2a}\sin(ax)\cos(ax)$$
I assume that a = $k\omega_0$ in my case.
When I try to work out the definite integral, when t = 0:$$b_k\frac 02 -\frac1{2k\omega_0}\sin(k\omega_00)\cos(k\omega_00) = 0$$
And when t = $T_0$:
$$b_k\frac {T_0}2 -\frac1{2k\omega_0}\sin(k\omega_0{T_0})\cos(k\omega_0{T_0})$$
Apart from knowing that the a sine and cosine integrated across a full period, $T_0$, is zero, I'm a bit confused as to how the term to the right of the subtraction sign "disappears" to give the neat answer shown in the textbook. Maybe the answer is self-evident from this knowledge, but I'm not sure if I've missed something else. Any help would be appreciated.
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