Question

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

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

(c) $-\frac{3}{5}$

(d) $-\frac{5}{3}$

Answer 1

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

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