Laplace transform of cosine squared function

[ChiralSuperfields]

Homework Statement

Please see below

Relevant Equations

##L[(\cos^2 (2t)] = L[\cos 2t] * L[\cos 2t]##

For part (b),

I have tried finding the Laplace transform of via the convolution property of Laplace transform.

My working is,

##L[\cos^2 (2t)] = L[\cos 2t] * L[\cos 2t]##
##L[\cos^2 (2t)] = \frac{s}{s^2 + 4} * \frac{s}{s^2 + 4}##
##\int_0^t \frac{s^2}{(s^2 + 4)^2} dt = \frac{ts^2}{(s^2 + 4)^2}##

However, I don't see how that is equivalent/equal to the expression they got for (b). Does some please know how or if I've made a mistake?

Thanks!

 

[Orodruin]

Your working is wrong. You are trying to make a convolution, but that is not how it is done. Please look up the actual expression for a convolution.

Nb: the easiest way to solve this is using trig identities as done in the solution.