Is 0.5sin2x the Fourier Series of sinx*cosx?

[fiziksfun]

My professor said the Fourier series of sinx*cosx is simply 0.5sin2x.

Is this right?? Is there no summation??

I don't understand why .. Help!

 

[Billy Bob]

She's right, by uniqueness of Fourier series.

Trig identity for sin2x shows that sin x*cos x=0.5sin2x.

This is the form of a Fourier series, hence the unique. (All the other infinitely many terms are 0.)