Question

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

Answer 1

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

We know that, $cot\theta =\frac{adjacentside}{oppositeside}=\frac{3}{4}$ [given]

Now,
$\Rightarrow Hypotenus{e}^2=adjacentsid{e}^2+oppositesid{e}^2\Rightarrow Hypotenus{e}^2=3^2+4^2\Rightarrow Hypotenus{e}^2=25\Rightarrow Hypotenuse=5$

$sec\theta =\frac{hypotenuse}{adjacentside}=\frac{5}{3}cosec\theta =\frac{hypotenuse}{oppositeside}=\frac{5}{4}$

Now, we will put values i.e.

$=\sqrt{(}\frac{sec\theta -cosec\theta }{sec\theta +cosec\theta })=\sqrt{(}\frac{\frac{5}{3}-\frac{5}{4}}{\frac{5}{3}+\frac{5}{4}})=\frac{\frac{5}{12}}{\frac{35}{12}}=\frac{5}{35}=\frac{1}{\surd 7}$

Hence, proved.