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प्रश्न
If \[cosec x - \cot x = \frac{1}{2}, 0 < x < \frac{\pi}{2},\]
पर्याय
- \[\frac{5}{3}\]
- \[\frac{3}{5}\]
- \[- \frac{3}{5}\]
- \[- \frac{5}{3}\]
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उत्तर
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 ):
\[2\text{ cosec} x = \frac{1}{2} + 2\]
\[ \Rightarrow 2 \text{ cosec} x = \frac{5}{2}\]
\[ \Rightarrow\text{ cosec} x = \frac{5}{4}\]
\[ \Rightarrow \frac{1}{\sin x}=\frac{5}{4}\]
\[ \Rightarrow \sin x=\frac{4}{5}\]
\[\text{ Now, }0 < \theta < \frac{\pi}{2}\]
\[ \therefore \cos\theta = \sqrt{1 - \sin^2 \theta}\]
\[ = \sqrt{1 - \left( \frac{4}{5} \right)^2}\]
\[ = \frac{3}{5}\]
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