Help in signals and fourier series

[brad sue]

Hi,
I have a question about Fourier Series(FS) in my textbook which is persentted like this:

The uniqueness of a FS means that if we can find the FS of a waveform, we are assured that there is no other waveform with that FS, except for waveforms differing from the waveform under consideration only over an inconsequential set of values of the independent variable. With this assitance, find the following trigonometric FS without doing any integration:

x(t)=cos^3(20*\pi*t)*[1-sin^2(10*\pi*t)]
The solution to this is supposed to be:

x(t)=\frac{5}{8}*cos(20*\pi*t)+\frac{5}{16}*cos(60*\pi*t)+\frac{1}{16}*cos(100*\pi*t)
all the powers have been eliminated.

I tried to use the Euler theorem and the trigonometric identity , but I could not find the solution.
When I expand using euler and trigo identity, I get 'frequencies' of 60*pi, 40*pi, 80*pi but not 100*pi as the solution suggests. I don't know if it is a typo .

Please can someone help me?
Thank you
B

 

[BackEMF]

Yep. I'd say you are correct. Can you post up your full answer & I'll just check if it looks what I think it should look like?

 

[brad sue]

Ok I found :
x(t)=cos(60*\pi*t)+\frac{1}{2}*cos(80*\pi*t)+\frac{1}{2}*cos(40*\pi*t)+3*cos(20*\pi*t)+\frac{3}{2}