Question

If $\text{cosec}x-\cot x=2-\sqrt{3}$, then what is the value of $\text{cosec}x+\cot x$?

• A. $\sqrt{3}-2$
• B. $\sqrt{3}+2$
• C. $2+\sqrt{3}$
• D. $2-\sqrt{3}$

Answer 1

The correct option is C $2+\sqrt{3}$

We know that ${\text{cosec}}^{2}x-{\cot }^{2}x=1$
$\Rightarrow (\text{cosec}x+\cot x)(\text{cosec}x-\cot x)=1$
$\Rightarrow \text{cosec}x+\cot x=\frac{1}{(\text{cosec}x-\cot x)}$

$\Rightarrow \text{cosec}x+\cot x=\frac{1}{2-\sqrt{3}}$
Now, Rationalizing the denominator:

$\Rightarrow \text{cosec}x+\cot x=\frac{1}{2-\sqrt{3}}\times \frac{2+\sqrt{3}}{2+\sqrt{3}}$

$\Rightarrow \text{cosec}x+\cot x=\frac{2+\sqrt{3}}{4-3}=2+\sqrt{3}$

$\Rightarrow \text{cosec}x+\cot x=2+\sqrt{3}$