Decompose the signal (1+0.1cos5t)cos100t into a linear combination of
sinusoidal function, and find the amplitude, frequency and phase of
each component. (Hint: Use the identity for cos a cos b).

Hi,

Thank you very much for the interesting question. 

The give signal can be decomposed into the following components:

=> (1 + 0.1cos5t)cos100t

=> cos100t + 0.1cos5tcos100t

Using the formula: cosAcosB = 1/2{cos(A+B) + cos(A-B)} we get,

=> cos100t + (0.1/2)cos105t  + (0.1/2)cos95t

=> cos100t + (0.05)cos105t + 0.05cos95t

Now the above is the decomposed signal into its different components.

Accordingly,

=> at 100/(2*pi)  = 50/pi Hz (~16Hz) The amplitude is 1(one).

=> at 105/(2*pi)  = 52.5/pi Hz (~17Hz) The amplitude is 0.05.

=> at 95/(2*pi)   = 47.5/pi Hz (~15 Hz) The amplitude is 0.05.

The phase difference in all of the above cases remains zero as there
is no other term within  the cos function.

Notes:

1.The decomposition of the given signal does not contain any sine
function terms which anyway can be readily obtained by appropriate
conversions.

2.The 'j' term (imaginary part) does not exist in this case.

3. The amplitude at all other frequencies except for the ones listed
above are zero.

If you have any doubts please do not hesitate to clarify before rating
this answer.

Thank you once again,

Warmest Regards,
Shiv Reddy

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