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In triangle ABC , with the usual notatio...

In `triangle ABC` , with the usual notations , prove that `2{a sin^(2).(C)/(2)+csin^2.(A)/(2)}=a+c-b`

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To prove the equation \( 2 \left( a \sin^2 \left( \frac{C}{2} \right) + c \sin^2 \left( \frac{A}{2} \right) \right) = a + c - b \) in triangle \( ABC \), we will follow these steps: ### Step 1: Rewrite the sine squared terms Using the identity \( \sin^2 \theta = \frac{1 - \cos 2\theta}{2} \), we can rewrite the sine squared terms in the left-hand side (LHS): \[ \sin^2 \left( \frac{C}{2} \right) = \frac{1 - \cos C}{2} \] ...
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