In `DeltaABC`, prove that:
`(a+b+c)(tanA/2+tanC/2)=2bcotB/2`

Explanation

To prove the equation \((a+b+c)\left(\tan\frac{A}{2} + \tan\frac{C}{2}\right) = 2b \cot\frac{B}{2}\) in triangle \(ABC\), we will follow these steps:

### Step 1: Express \(\tan\frac{A}{2}\) and \(\tan\frac{C}{2}\) in terms of sides of the triangle
Using the formula for \(\tan\frac{A}{2}\) and \(\tan\frac{C}{2}\):
\[ \tan\frac{A}{2} = \frac{\sqrt{s(s-a)}}{s} \quad \text{and} \quad \tan\frac{C}{2} = \frac{\sqrt{s(s-c)}}{s} \]
where \(s = \frac{a+b+c}{2}\) is the semi-perimeter of the triangle.
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