If `t a ntheta/2=sqrt((a-b)/(a+b))tanvarphi/2` , prove that `costheta=(a cosvarphi+b)/(a+b cosvarphi)` .

Explanation

To prove that \( \cos \theta = \frac{a \cos \varphi + b}{a + b \cos \varphi} \) given that \( \tan \frac{\theta}{2} = \sqrt{\frac{a-b}{a+b}} \tan \frac{\varphi}{2} \), we will start from the left-hand side and simplify it step by step.

### Step 1: Use the half-angle identity for cosine
We know that:
\[ \cos \theta = \frac{1 - \tan^2 \frac{\theta}{2}}{1 + \tan^2 \frac{\theta}{2}} \]
Substituting the given expression for \( \tan \frac{\theta}{2} \):
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