Question

if b cos thetha =a,prove cosec thetha+cot thetha
=√b+a/√b-a​

Answer 1

SOLUTION:-

Given,

$b \: cos \theta = a \\ = > cos \theta = \frac{a}{b}$

Now,

$sin \theta = \sqrt{1 - {cos}^{2} \theta } = \sqrt{1 - \frac{ {a}^{2} }{ {b}^{2} } } = \frac{ \sqrt{ {b}^{2} - {a}^{2} } }{b}$

Now,

$cosec \theta = \frac{1}{sin \theta} = \frac{b}{ \sqrt{ {b}^{2} - {a}^{2} } } \\ \\ cot \theta = \frac{cos \theta}{sin \theta} = \frac{ \frac{a}{b} }{ \sqrt{ {b}^{2} - \frac{ {a}^{2} }{b} } } = \frac{a}{ \sqrt{ {b}^{2} - {a}^{2} } } \\ \\ cosec \theta + cot \theta = \frac{b}{ \sqrt{ {b}^{2} - {a}^{2} } } + \frac{a}{ \sqrt{ {b}^{2} - {a}^{2} } } = \frac{b + a}{ \sqrt{ {b}^{2} - {a}^{2} } } \\ \\ = > \frac{(b + a)}{ \sqrt{b + a} \sqrt{b + a} } \\ \\ = > \frac{ \sqrt{b + a} }{ \sqrt{b - a} }$

Proved.

Hope it helps ☺️