If the sides `a ,b ,c` of a `triangleABC` are in `HP,` prove that `sin^2(A/2),sin^2(B/2),sin^2(C/2)` are in `HP`

Explanation

To prove that if the sides \( a, b, c \) of triangle \( ABC \) are in Harmonic Progression (HP), then \( \sin^2\left(\frac{A}{2}\right), \sin^2\left(\frac{B}{2}\right), \sin^2\left(\frac{C}{2}\right) \) are also in HP, we can follow these steps:

### Step 1: Understanding Harmonic Progression
If the sides \( a, b, c \) are in HP, then their reciprocals \( \frac{1}{a}, \frac{1}{b}, \frac{1}{c} \) are in Arithmetic Progression (AP). This means:
\[ 2 \cdot \frac{1}{b} = \frac{1}{a} + \frac{1}{c} \]
or equivalently,
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