How to find the Laplace Transform of two (independent) functions multiplied together?

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How does one find the laplace transform for an equation consisting of two trig functions multiplied together, when it is not possible to use any trig identities? For example, take a function say;

f(t)=cos(t)*sin(5t)

As far as I am aware, there isn't any trig identity to "convert" such an equation to consist of only one trig function, and to thus be able to do the laplace transform. Or am I missing some fine detail/trick here? Thanks in advance for any help you may provide!

The only other way I can think of is to do the actual integral to find the Laplace transform. So I guess, I should ask if that is the only way it is possible to do as such?