Prove that:`(sinx+sin3x)/(cosx+cos3x)=tan2x`

Explanation

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

### Step-by-step Solution:

1. **Start with the LHS:**
\[ \text{LHS} = \frac{\sin x + \sin 3x}{\cos x + \cos 3x} \]
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