Question : If $\theta$ is a positive acute angles and $\operatorname{cosec}\theta =\sqrt{3}$, then the value of $\cot \theta -\operatorname{cosec}\theta$ is:
Option 1: $\sqrt2-\sqrt3$
Option 2: $\frac{\sqrt{2}(3+\sqrt{3})}{3}$
Option 3: $\frac{\sqrt{2}(3-\sqrt{3})}{3}$
Option 4: $\frac{3\sqrt{2}+\sqrt{3}}{3}$
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Correct Answer: $\sqrt2-\sqrt3$
Solution :
Given: $\operatorname{cosec}\theta=\sqrt3$
$⇒\frac{1}{\sin\theta}=\sqrt3$
$⇒\sin\theta=\frac{1}{\sqrt3}$
We know that,
$\cos\theta = \sqrt{1-\sin^2\theta}=\sqrt{1-(\frac{1}{\sqrt3})^2}=\sqrt{1-\frac{1}{3}}=\sqrt\frac{2}{3}$
So, $\cot\theta-\operatorname{cosec}\theta$
$=\frac{\cos\theta}{\sin\theta}-\operatorname{cosec}\theta$
$=\frac{\sqrt\frac{2}{3}}{\frac{1}{\sqrt3}}-\sqrt3$
$=\sqrt\frac{2}{3}×\sqrt3-\sqrt3$
$=\sqrt2-\sqrt3$
Hence, the correct answer is $\sqrt2-\sqrt3$.
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