`(sin5x+sin3x)/(cos5x+cos3x)=tan4x`

Explanation

To prove that \(\frac{\sin 5x + \sin 3x}{\cos 5x + \cos 3x} = \tan 4x\), we will use the sum-to-product identities for sine and cosine.

### Step 1: Apply the sum-to-product identities for sine
We know that:
\[ \sin A + \sin B = 2 \sin\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \]
Let \(A = 5x\) and \(B = 3x\). Then:
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