Question

If $\cos \theta =\frac{3}{4}$ then the value of $\sqrt{\frac{\sec \theta -cosec\theta }{\sec \theta +cosec\theta }}$ is ____

Answer 1

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

$\cos \theta =\frac{3}{4}=\frac{b}{h}$

$p=\sqrt{h^2-b^2}=\sqrt{4^2-3^2}$
$p=\sqrt{4^2-3^2}=\sqrt{16-9}=\sqrt{7}$
$cosec\theta =\frac{h}{p}=\frac{4}{\sqrt{7}}$
$sec\theta =\frac{h}{b}=\frac{4}{3}$
$\sqrt{\frac{\sec \theta -cosec\theta }{\sec \theta +cosec\theta }}=\sqrt{\frac{\frac{4}{3}-\frac{4}{\sqrt{7}}}{\frac{4}{3}+\frac{4}{\sqrt{7}}}}$

$\sqrt{\frac{\sec \theta -cosec\theta }{\sec \theta +cosec\theta }}=\sqrt{\frac{\frac{4(\sqrt{7}-3)}{3\sqrt{7}}}{\frac{4(\sqrt{7}+3)}{3\sqrt{7}}}}$