use partial fractions for the inverse transform
$\sin^2(t)=\frac{1}{2} (1-\cos (2 t))$
you should be able to finish from there.
Part A) This is the product of two Laplace transforms (Hint: Convolution Theorem is of interest here) :
What are and ? What are the corresponding and ?
Part B) Laplace transform of Recall that and we also have the definition of the Laplace transform: . What do you get when you substitute the above trig identity in the definition of the laplace transform (Hint: you should notice the answer will be a combination of two simpler Laplace transforms)