Prove that:`(sin5x+sin3x)/(cos5x+cos3x)=tan4x`

Explanation

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

### Step-by-Step Solution:

1. **Start with the left-hand side (LHS)**:
\[ \text{LHS} = \frac{\sin 5x + \sin 3x}{\cos 5x + \cos 3x} \]
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