Question

$\mathrm{If}\cos \theta =\frac{2}{3},\mathrm{then}\mathrm{write}\mathrm{the}\mathrm{value}\mathrm{of}\frac{\left(\sec \theta -1\right)}{\left(\sec \theta +1\right)}.$

Answer 1

$$\begin{gathered} \frac{\sec \theta -1}{\sec \theta +1} \\ =\frac{\left({\displaystyle \frac{1}{\cos \theta }}-{\displaystyle \frac{1}{1}}\right)}{\left({\displaystyle \frac{1}{\cos \theta }}+{\displaystyle \frac{1}{1}}\right)} \\ =\frac{\left({\displaystyle \frac{1-\cos \theta }{\cos \theta }}\right)}{\left({\displaystyle \frac{1+\cos \theta }{\cos \theta }}\right)} \\ =\frac{1-\cos \theta }{1+\cos \theta } \\ =\frac{\left({\displaystyle \frac{1}{1}}-{\displaystyle \frac{2}{3}}\right)}{\left({\displaystyle \frac{1}{1}}+{\displaystyle \frac{2}{3}}\right)} \\ =\frac{\left({\displaystyle \frac{1}{3}}\right)}{\left({\displaystyle \frac{5}{3}}\right)} \\ =\frac{1}{5} \end{gathered}$$