In a triangle $A B C$ , if $b = (\sqrt{3} - 1) a$ and $∠ C = 30^{o}$ , then the value of $(A - B)$ is equal to(All symbols used have usual meaning in a triangle.)

30^{o}

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45^{o}

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60^{o}

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75^{o}

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Solution

The correct option is A $30^{o}$
$\begin{matrix} \frac{b}{a} = \sqrt{3} - 1 ∠ c = 30^{\circ} \frac{b}{a} = \frac{sin B}{sin A} = \sqrt{3} - 1 \frac{sin (180 - (A + C))}{sin A} = \sqrt{3} - 1 \frac{sin (A + C)}{sin A} = \sqrt{3} - 1 \frac{sin A . cos C + sin C . cos A}{sin A} = \sqrt{3} - 1 cos 30 + sin 30. cot A = \sqrt{3} - 1 \frac{\sqrt{3}}{2} + \frac{1}{2} cot A = \sqrt{3} - 1 \frac{1}{2} cot A = \sqrt{3} - 1 - \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2} - 1 cot A = 2 (\frac{\sqrt{3}}{2} - 1) = \sqrt{3} - 2 \end{matrix}$

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