$If\ heta$ is an acute angle and $ an{ heta = \frac{1}{\sqrt{7}}}$, then the value of $\frac{{cosec}^{2} heta - \sec^{2} heta}{{cosec}^{2} heta + \sec^{2} heta}$ is

Question

$If\ 	heta$ is an acute angle and $	an{	heta = \frac{1}{\sqrt{7}}}$, then the value of $\frac{{cosec}^{2}	heta - \sec^{2}	heta}{{cosec}^{2}	heta + \sec^{2}	heta}$ is

Dr. Abubacker · Accepted Answer

(a)
Given, $	an{	heta = \frac{1}{\sqrt{7}}}$
$	herefore\ \frac{cosec^{2} - \sec^{2}	heta}{cosec^{2}	heta + \sec^{2}	heta} = \frac{\left( 1 + \cot^{2}	heta ight) - \left( 1 + 	an^{2}	heta ight)}{\left( 1 + \cot^{2}	heta ight) + \left( 1 + 	an^{2}	heta ight)}$
$= \frac{\left( \cot^{2}{	heta - 	an^{2}	heta} ight)}{2 + 	an^{2}{	heta + \cot^{2}	heta}}$
$= \frac{7 - \frac{1}{7}}{2 + \frac{1}{7} + 7} = \frac{48}{64} = \frac{3}{4}$

Q. $If\ \theta$ is an acute angle and $\tan{\theta = \frac{1}{\sqrt{7}}}$, then the value of $\frac{{cosec}^{2}\theta - \sec^{2}\theta}{{cosec}^{2}\theta + \sec^{2}\theta}$ is

Explanation

Reviewed by Dr. Abubacker

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