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Sum of n Terms

If A, B, ...

Question

If $A$ , $B$ , $C$ be the angles of a triangle and
$∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 & 1 1 + sin A & 1 + sin B & 1 + sin C sin A + {sin}^{2} A & sin B + {sin}^{2} B & sin C + {sin}^{2} C \end{matrix} ∣ ∣ ∣ ∣ ∣ = 0$
then prove that $Δ$ must be isosceles.

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Solution

Apply $R_{2} - R_{1}$ and then $R_{3} - R_{2}$ (new).
$Δ = (sin A - sin B) (sin B - sin C) (sin C - sin A) = 0$
Hence $Δ$ is isosceles as in part $(b)$ .

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