If by laplace of a product of functions you mean something such as L(t sin at), then there are several methods.
(1) We could differentiate the product of functions are make use of the following laplace property:
L(f'') = s^2 L(f) - sf(0) - f'(0)
Let f(t) = t sin at
f'(t) = at cos at + sin at
f''(t) = -a^2 t sin at + a cos at
L(f''(t)) = -a^2 L(t sin at) + a L(cos at)
= s^2 L(t sin at) - s (0 sin 0) - (0 + sin 0)
= s^2 L(t sin at)
Rearranging, we get
(s^2 + a^2) L(t sin at) = aL(cos at)
Which you should be able to solve since you know L(cos at)
(2) Or you could go with the definition of the Laplace transform.
First start with L(sin at) = a/ (a^2 + s^2) = integrate (e^-st . (sin at)) dt
differentiate both the Left hand side, and right hand side, with respect to s
differentiate (a/ (a^2 + s^2)) = integrate (-te^-st . (sin at)) dt = - integrate (e^-st . tsin at) dt which by definition is L(t sin at)
