Question

If $cosec\theta -cot\theta =\frac{1}{2},0<\theta <\frac{\pi }{2}$, then $cos\theta$ is equal to

• A. $\frac{5}{3}$
• B. $\frac{3}{5}$
• C. $-\frac{3}{5}$
• D. $-\frac{5}{3}$

Answer 1

The correct option is B $\frac{3}{5}$

Given: $cosec\theta -cot\theta =\frac{1}{2}.....\left(1\right)\Rightarrow cosec\theta +cot\theta =\mathrm{2.....}\left(2\right)(\text{since,}cose{c}^2\theta -co{t}^2\theta =1)\text{Adding (1) and (2)},2cosec\theta =\frac{1}{2}+2=\frac{5}{2}\Rightarrow cosec\theta =\frac{5}{4}\Rightarrow sin\theta =\frac{4}{5}\Rightarrow si{n}^2\theta =\frac{16}{25}\Rightarrow 1-co{s}^2\theta =\frac{16}{25}\Rightarrow co{s}^2\theta =1-\frac{16}{25}=\frac{9}{25}\Rightarrow cos\theta =\frac{3}{5}$