Question : If $\sin A=\frac{\sqrt{3}}{2}, 0<A<90^{\circ}$, then find the value of $2(\operatorname{cosec} A + \cot A)$.
Option 1: $2 \sqrt{3}$
Option 2: $\sqrt{3}$
Option 3: $\frac{2}{\sqrt{3}}$
Option 4: $\frac{1}{\sqrt{3}}$
Correct Answer: $2 \sqrt{3}$
Solution :
Given:
$\sin A\ =\ \frac{\sqrt{3}}{2}=\sin 60^\circ$
$\Rightarrow A\ =\ 60^\circ$
Now, putting the value of A, we get,
$2(\operatorname{cosec} A + \cot A)$
$=\ 2(\operatorname{cosec} 60^\circ + \cot 60^\circ)$
$=\ 2(\frac{2}{\sqrt{3}}\ +\ \frac{1}{\sqrt{3}})$
$=\ 2(\frac{3}{\sqrt{3}})$
$=\ 2\sqrt{3}$
Hence the value of expression is $2\sqrt{3}$.
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