If in `Delta ABC, tan.(A)/(2) = (5)/(6) and tan.( C)/(2)= (2)/(5)`, then prove that a, b, and c are in A.P.

Explanation

To prove that angles A, B, and C of triangle ABC are in Arithmetic Progression (A.P.), given that \( \tan\left(\frac{A}{2}\right) = \frac{5}{6} \) and \( \tan\left(\frac{C}{2}\right) = \frac{2}{5} \), we will follow these steps:

### Step 1: Use the Half-Angle Tangent Formula
We know that in a triangle, the tangent of half an angle can be expressed using the area (Δ) and the semi-perimeter (S):
\[ \tan\left(\frac{A}{2}\right) = \frac{\Delta}{S(S - A)} \]
\[
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