Question

If \[cosec x - \cot x = \frac{1}{2}, 0 < x < \frac{\pi}{2},\]

 

Options

• \[\frac{5}{3}\]

   

• \[\frac{3}{5}\]

   

• \[- \frac{3}{5}\]

   

• \[- \frac{5}{3}\]

   

Answer 1

\[\frac{3}{5}\]

We have: 

\[\text{ cosec }x - \cot x = \frac{1}{2} \left( 1 \right)\]

\[ \Rightarrow \frac{1}{\text{ cosec }x - \cot x} = 2\]

\[ \Rightarrow \frac{{\text{ cosec }}^2 x - \cot^2 x}{\text{ cosec }x - \cot x} = 2\]

\[ \Rightarrow \frac{\left(\text{ cosec }x + \cot x \right)\left( \text{ cosec }x - \cot x \right)}{\left(\text{ cosec }x - \cot x \right)} = 2\]

\[ \therefore\text{ cosec }x +\cot x = 2 \left( 2 \right)\]

Adding ( 1 ) and ( 2 ): 

\[2\text{ cosec} x = \frac{1}{2} + 2\]

\[ \Rightarrow 2 \text{ cosec} x = \frac{5}{2}\]

\[ \Rightarrow\text{ cosec} x = \frac{5}{4}\]

\[ \Rightarrow \frac{1}{\sin x}=\frac{5}{4}\]

\[ \Rightarrow \sin x=\frac{4}{5}\]
\[\text{ Now, }0 < \theta < \frac{\pi}{2}\]
\[ \therefore \cos\theta = \sqrt{1 - \sin^2 \theta}\]
\[ = \sqrt{1 - \left( \frac{4}{5} \right)^2}\]
\[ = \frac{3}{5}\]