This is part of a solution but I don't understand the underlined part. Why is it equals to 0?
sin(wt)/6 x sin(2wt-(pi/4))
is not equal to zero.
That term itself isn't zero. But the area under it, taken over the complete cycle 0→2 Pi contributes nothing to the integral that you are about to evaluate. So the author is just looking ahead and seeing that he can save himself a bit of work here, and not bothering to process a term that is going to end up as zero anyway.
Why don't you work through without dismissing that term, and see how the result pans out?
An easy way to show that that term integrates to zero is to use the identity ##\sin(a)\sin(b) = \frac 1 2 \bigl(\cos(a-b) - \cos(a+b)\bigr)##.
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