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Implicit Differentiation

If A+B+C=π,...

Question

If $A + B + C = π$ , then prove that $cot A + \frac{sin A}{sin B sin C}$ retains the same value when any two of the angles A, B, C are interchanged.

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Solution

$\frac{sin A}{sin B sin C} = \frac{sin (B + C)}{sin B sin C} = cot B + cot C$
$∴$ Given expression $= \sum cot A$ . Hence proved by symmetry.

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