Question

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

Answer 1

Given,

$\cot \theta =\frac{3}{4}$

$\Rightarrow \tan \theta =\frac{4}{3}$

$\Rightarrow \sin \theta =\frac{4}{5},\cos \theta =\frac{3}{5}$

$\csc \theta =\frac{5}{4},\sec \theta =\frac{5}{3}$

from Pythagorean triplets

Given,

$\sqrt{\frac{\sec \theta -\csc \theta }{\sec \theta +\csc \theta }}$

$=\sqrt{\frac{\frac{5}{3}-\frac{5}{4}}{\frac{5}{3}+\frac{5}{4}}}$

$=\sqrt{\frac{\frac{5}{12}}{\frac{12}{35}}}$

$=\frac{1}{\sqrt{7}}$