If `2t a nalpha/2=tan(beta/2)` , prove that `cosalpha=(3+5cosbeta)/(5+3cosbeta)`

Explanation

To prove that \( \cos \alpha = \frac{3 + 5 \cos \beta}{5 + 3 \cos \beta} \) given that \( 2 \tan \frac{\alpha}{2} = \tan \frac{\beta}{2} \), we can follow these steps:

### Step 1: Express \( \cos \alpha \) in terms of \( \tan \frac{\alpha}{2} \)

Using the formula for cosine in terms of tangent half-angle:
\[ \cos \alpha = \frac{1 - \tan^2 \frac{\alpha}{2}}{1 + \tan^2 \frac{\alpha}{2}} \]
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