Prove that: `sin 3x + sin 2x + sin x = 4 sin x cos (x/2) cos ((3x)/2)`

Explanation

To prove the identity \( \sin 3x + \sin 2x + \sin x = 4 \sin x \cos \left( \frac{x}{2} \right) \cos \left( \frac{3x}{2} \right) \), we will start with the left-hand side (LHS) and manipulate it to match the right-hand side (RHS).

### Step 1: Write the LHS
We start with the left-hand side:
\[ \text{LHS} = \sin 3x + \sin 2x + \sin x \]

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