The image contains a problem related to the Laplace transform, denoted as $L$. The problem statement and parts of the solution seem to be about proving certain properties of the Laplace transform and then using those properties to evaluate specific functions.

Problem Description:

Given that $L[f(t)] = F(s)$, the problem asks to show two specific properties of the Laplace transform:

1. $L[\sin(at)f(t)] = \frac{1}{2i} \left[F(s - ai) - F(s + ai)\right]$
2. $L[\cos(at)f(t)] = \frac{1}{2} \left[F(s - ai) + F(s + ai)\right]$

Then, the problem asks to evaluate:

(i) $L[\sin(2t)\sin(3t)]$

(ii) $L[\cos(3t)\cos(2t)]$

Steps to Solve:

1. Show the properties:

   • For $L[\sin(at)f(t)]$, using the definition of the Laplace transform and some trigonometric identities, the proof can be derived.
   • For $L[\cos(at)f(t)]$, a similar process is used, with the final result expressed as a sum rather than a difference.

2. Evaluate the given Laplace Transforms:

   • Use the identities proven in the first part to evaluate the specific Laplace transforms for the given trigonometric products.

Would you like a detailed step-by-step solution for each part of this problem?