Prove that `cos 2xcos(x/2)-cos 3x cos((9x)/2)= sin5x sin((5x)/2)`.

Explanation

To prove that \( \cos(2x) \cos\left(\frac{x}{2}\right) - \cos(3x) \cos\left(\frac{9x}{2}\right) = \sin(5x) \sin\left(\frac{5x}{2}\right) \), we will use trigonometric identities.

### Step 1: Apply the product-to-sum identities
We start with the left-hand side of the equation:
\[ \cos(2x) \cos\left(\frac{x}{2}\right) - \cos(3x) \cos\left(\frac{9x}{2}\right) \]

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