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

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

I tried to use the Euler theorem and the trigonometric identity , but I could not find the solution.

Please can someone help me?
Thank you