If `tan beta=cos theta tan alpha`, then prove that `tan^(2)""(theta)/(2)=(sin(alpha-beta))/(sin(alpha+beta))`.

Explanation

To prove that \( \frac{\tan^2 \theta}{2} = \frac{\sin(\alpha - \beta)}{\sin(\alpha + \beta)} \) given that \( \tan \beta = \cos \theta \tan \alpha \), we will start with the left-hand side (LHS) and manipulate it step by step.

### Step 1: Start with LHS
We begin with the left-hand side:
\[ \frac{\tan^2 \theta}{2} \]

...