Determine the Laplace transform of cos^2 t
We can use the trigonometric identity cos^2 t = (1 + cos 2t)/2 to write:
L{cos^2 t} = L{(1 + cos 2t)/2}
Using the linearity property of the Laplace transform, we can split this into two parts:
L{cos^2 t} = 1/2 * L{1} + 1/2 * L{cos 2t}
The Laplace transform of a constant function 1 is simply 1/s. For the second part, we can use the formula L{cos at} = s/(s^2 + a^2), with a = 2, to get:
L{cos 2t} = s/(s^2 + 2^2) = s/(s^2 + 4)
Putting everything together, we get:
L{cos^2 t} = 1/2 * 1/s + 1/2 * s/(s^2 + 4)
= (s^2 + 2)/(2s(s^2 + 4))