Question

If $\cot \theta =\frac{3}{4}$, show that $\sqrt{\frac{\sec \theta -\mathrm{cosec}\theta }{\sec \theta +\mathrm{cosec}\theta }}=\frac{1}{\sqrt{7}}$.

Answer 1

$$\begin{gathered} \mathrm{LHS}=\sqrt{\frac{\sec \theta -\mathrm{cosec}\theta }{\sec \theta +\mathrm{cosec}\theta }} \\ =\sqrt{\frac{\left({\displaystyle \frac{1}{\cos \theta }}-{\displaystyle \frac{1}{\sin \theta }}\right)}{\left({\displaystyle \frac{1}{\cos \theta }}+{\displaystyle \frac{1}{\sin \theta }}\right)}} \\ =\sqrt{\frac{\left({\displaystyle \frac{\sin \theta -\cos \theta }{\sin \theta \cos \theta }}\right)}{\left({\displaystyle \frac{\sin \theta +\cos \theta }{\sin \theta \cos \theta }}\right)}} \\ =\sqrt{\frac{\left({\displaystyle \frac{\sin \theta -\cos \theta }{\sin \theta }}\right)}{\left({\displaystyle \frac{\sin \theta +\cos \theta }{\sin \theta }}\right)}} \\ =\sqrt{\frac{\left({\displaystyle \frac{\sin \theta }{\sin \theta }-\frac{{\displaystyle \cos \theta }}{\sin \theta }}\right)}{\left({\displaystyle \frac{\sin \theta }{\sin \theta }}+{\displaystyle \frac{\cos \theta }{\sin \theta }}\right)}} \\ =\sqrt{\frac{1-\cot \theta }{1+\cot \theta }} \\ =\sqrt{\frac{\left(1-{\displaystyle \frac{3}{4}}\right)}{\left(1+{\displaystyle \frac{3}{4}}\right)}} \end{gathered}$$
$$\begin{gathered} =\sqrt{\frac{\left({\displaystyle \frac{1}{4}}\right)}{\left({\displaystyle \frac{7}{4}}\right)}} \\ =\sqrt{\frac{1}{7}} \\ =\frac{1}{\sqrt{7}} \\ =\mathrm{RHS} \end{gathered}$$