Question : If $\operatorname{cosec B} = \frac{3}{2}$, then what is the value of $\mathrm{\cot B \sin B} $?
Option 1: $\frac{\sqrt{5}}{3}$
Option 2: $\frac{4}{3 \sqrt{3}}$
Option 3: $\frac{3 \sqrt{2}}{2}$
Option 4: $\frac{2 \sqrt{5}}{3}$
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Correct Answer: $\frac{\sqrt{5}}{3}$
Solution :
Given that $\operatorname{cosec B} = \frac{3}{2}$.
$\mathrm{\sin B = \frac{2}{3}}$
$⇒\mathrm{\cos B = \sqrt{1 - \sin^2 B} = \sqrt{1 - \left(\frac{2}{3}\right)^2} = \sqrt{1 - \frac{4}{9}} = \sqrt{\frac{5}{9}} = \frac{\sqrt{5}}{3}}$
$⇒\mathrm{\cot B = \frac{\cos B}{\sin B} = \frac{\frac{\sqrt{5}}{3}}{\frac{2}{3}} = \frac{\sqrt{5}}{2}}$
$\therefore \mathrm{\cot B \sin B = \frac{\sqrt{5}}{2} \times \frac{2}{3} = \frac{\sqrt{5}}{3}}$
Hence, the correct answer is $\frac{\sqrt{5}}{3}$.
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