Question

say the answer with step by step explanation​

Answer 1

Step-by-step explanation:

√1+cos theta / 1-cos theta * √1+cos theta / 1+cos theta

√ (1+cos theta)^2/ 1- cos ^2 theta

√(1+cos theta)^2/ sin ^2 theta

1+ cos theta / sin theta

1/ sin theta + cos theta / sin theta

cosec theta + cot theta

Answer 2

Required Answer:-

Given to Prove:

$\rm \mapsto \sqrt{ \dfrac{1 + \cos(x) }{1 - \cos(x) } } = cosec(x) + \cot(x)$

Proof:

Here comes the proof.

Taking LHS,

$\rm \sqrt{ \dfrac{1 + \cos(x) }{1 - \cos(x) } }$

$\rm = \sqrt{ \dfrac{(1 + \cos(x))(1 + \cos(x)) }{(1 - \cos(x))(1 + \cos(x)) } }$

$\rm = \sqrt{ \dfrac{(1 + \cos(x))(1 + \cos(x)) }{ {(1)}^{2} - { \cos}^{2}(x) }}$

We know that,

$\rm \implies { \sin}^{2}(x) + \cos ^{2} (x) = 1$

$\rm \implies { \sin}^{2}(x) = 1 - \cos ^{2} (x)$

Therefore,

$\rm \sqrt{ \dfrac{(1 + \cos(x))(1 + \cos(x)) }{ {(1)}^{2} - { \cos}^{2}(x) }}$

$\rm = \sqrt{ \dfrac{(1 + \cos(x))(1 + \cos(x)) }{ { \sin}^{2}(x) }}$

$\rm = \sqrt{ \dfrac{(1 + \cos(x))^{2} }{ { \sin}^{2}(x) }}$

$\rm = \sqrt{ \bigg(\dfrac{1 + \cos(x) }{\sin(x) } \bigg)^{2} }$

$\rm = \bigg(\dfrac{1 + \cos(x) }{\sin(x) } \bigg)$

$\rm = \dfrac{1 }{\sin(x) } + \dfrac{ \cos(x) }{ \sin(x) }$

As we know that,

• cosec(x) = 1/sin(x) and,
• tan(x) = sin(x)/cos(x)

So,

$\rm \dfrac{1 }{\sin(x) } + \dfrac{ \cos(x) }{ \sin(x) }$

$\rm =cosec(x) + \dfrac{1}{ \tan(x) }$

Also, cot(x) = 1/tan(x), Thus,

$\rm =cosec(x) + \cot(x)$

= RHS (Hence Proved)