Question

Solve : $\cot x+\text{cosec}x=\sqrt{3}$

Answer 1

$\cot x+\csc x=\sqrt{3}$...(1)

We know that ,

${\csc }^{2}x-{\cot }^{2}x=1$

$(\csc x+\cot x)(\csc x-\cot x)=1$

$\sqrt{3}(\csc x-\cot x)=1$ from ( 1 )

$\csc x-\cot x=\frac{1}{\sqrt{3}}$....(2)

Add equations ( 1 ) and ( 2 ) we get,

$2\csc x=\sqrt{3}+\frac{1}{\sqrt{3}}$

$=\frac{3+1}{\sqrt{3}}$

$=\frac{4}{\sqrt{3}}$

Therefore,

$\csc x=\frac{4}{2\sqrt{3}}$

$\csc x=\frac{2}{\sqrt{3}}$

$\csc x=\csc 60$°

$x=60$°