The difference between two value of $c$ is

2 \sqrt{(a^{2} - b^{2})}

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\sqrt{(a^{2} - b^{2})}

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2 \sqrt{(a^{2} - b^{2} {sin}^{2} A)}

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\sqrt{(a^{2} - b^{2} {sin}^{2} A)}

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Solution

The correct option is C $2 \sqrt{(a^{2} - b^{2} {sin}^{2} A)}$
$a, b, ∠ A$ are given $cos A = \frac{b^{2} + c^{2} - a^{2}}{2 b c}$
$\Rightarrow c^{2} - (2 b cos A) c + b^{2} - a^{2} = 0$
$\Rightarrow c = \frac{2 b cos A \pm \sqrt{4 b^{2} {cos}^{2} A - 4 (b^{2} - a^{2})}}{2}$
$\Rightarrow c = b cos A \pm \sqrt{a^{2} - b^{2} {sin}^{2} A}$
$| c_{1} - c_{2} | = 2 \sqrt{(a^{2} - b^{2} {sin}^{2} A)}$

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