Question

Prove that:
√1+cos A/√1-cosA
= cosecA+cotA.​

The question is from trigonometric identities from class 10.

Answer 1

Given

$\sf : \implies \sqrt{ \dfrac{1 + cosA}{1 - cosA} } = cosecA + cotA$

To Prove

$\sf : \implies \: cosecA + cotA$

Now Take

$\sf : \implies \sqrt{ \dfrac{1 + cosA}{1 - cosA} }$

Rationalize The Denominator

$\sf : \implies \sqrt{ \dfrac{1 + cosA}{1 - cosA} \times \dfrac{1 + cosA}{1 + cosA} }$

$\sf : \implies \: \sqrt{ \dfrac{(1 + cosA)^{2} }{(1 - cos ^{2}A) } }$

We Know that

$\sf : \implies \: sin {}^{2} x + cos {}^{2} x = 1$

$: \implies \sf \: sin^{2} x = 1 - cos {}^{2} x$

Put the value

$\sf : \implies \sqrt{ \dfrac{(1 + cosA) {}^{2} }{sin^{2} A} }$

$\sf : \implies \dfrac{1 + cosA}{sinA}$

$\sf : \implies \: \dfrac{1}{sinA} + \dfrac{cosA}{sinA}$

$\sf : \implies \: cosecA + cotA$

Hence Proved