If $sin A sin B sin C + cos A cos B = 1$ , then the value of $sin C =$ ______

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Solution

Given
$sin A sin B sin C + cos A cos B = 1$
$2 sin A sin B sin C + 2 cos A cos B = 2$
$\Rightarrow 2 sin A sin B sin C + 2 cos A cos B = ({sin}^{2} A + {cos}^{2} A) + ({sin}^{2} B + {cos}^{2} B)$
$\Rightarrow ({sin}^{2} A + {sin}^{2} B - 2 sin A sin B) + ({cos}^{2} A + {cos}^{2} B - 2 cos A cos B) - 2 sin A sin B sin C + 2 sin A sin B = 0$
$(sin A + sin B)^{2} + (cos A - cos B)^{2} + 2 sin A sin B (1 - sin C) = 0$
$∴ sin A - sin B = 0, 1 - sin C = 0$
$∴ sin A = sin B, \Rightarrow sin C = 1$
$∴ A = B$
Hence $sin C = 1$

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