Question

If $\mathrm{cosec\theta }-\mathrm{cot\theta }=\frac{1}{2},0<\mathrm{\theta }<\frac{\mathrm{\pi }}{2},$, then cos θ is equal to
(a) $\frac{5}{3}$
(b) $\frac{3}{5}$
(c) $-\frac{3}{5}$
(d) $-\frac{5}{3}$

Answer 1

(b) $\frac{3}{5}$

$$\begin{gathered} \mathrm{We}\mathrm{have}: \\ \mathrm{cosec}\theta -\cot \theta =\frac{1}{2}\left(1\right) \\ \Rightarrow \frac{1}{\mathrm{cosec}\theta -\cot \theta }=2 \\ \Rightarrow \frac{{\mathrm{cosec}}^2\theta -{\cot }^2\theta }{\mathrm{cosec}\theta -\cot \theta }=2 \\ \Rightarrow \frac{\left(\mathrm{cosec}\theta +\cot \theta \right)\left(\mathrm{cosec}\theta -\cot \theta \right)}{\left(\mathrm{cosec}\theta -\cot \theta \right)}=2 \\ \therefore \mathrm{c}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{c}\theta +\cot \theta =2\left(2\right) \\ \mathrm{Adding}\left(1\right)\mathrm{and}\left(2\right): \\ 2\cos \text{ec}\theta =\frac{1}{2}+2 \\ \Rightarrow 2\mathrm{cosec}\theta =\frac{5}{2} \\ \Rightarrow \mathrm{cosec}\theta =\frac{5}{4} \\ \Rightarrow \frac{1}{\sin \theta }=\frac{5}{4} \\ \Rightarrow \sin \theta =\frac{4}{5} \end{gathered}$$
$$\begin{gathered} \mathrm{Now},0<\theta <\frac{\mathrm{\pi }}{2} \\ \therefore \cos \theta =\sqrt{1-{\sin }^2\theta } \\ =\sqrt{1-{\left({\displaystyle \frac{4}{5}}\right)}^2} \\ =\frac{3}{5} \end{gathered}$$