Question

Prove it cos (a) -sin (a) +1 / cos (a) + sin (a) -1 = cosec ( a) + cot (a) ​

Answer 1

Step-by-step explanation:

We need to prove 1+cosA1−cosA=cosecA−cotA

By taking L.H.S. =1+cosA1−cosA

                           =1+cosA1−cosA×1−cosA1−cosA

                           =1−cos2A(1−cosA)2

                           =sin2A(1−cosA)2

                           =sinA1−cosA

                           =sinA1−sinAcosA

                           =cosecA−cotA= R.H.S.

Hope This will help you  

Answer 2

$\small\bold{\underline{\sf{\green{Given:-}}}}$

$\\ \implies { \bold{ \dfrac{ \cos(a) - \sin(a) + 1}{ \cos(a) + \sin(a) - 1} = cosec(a) + \cot(a) }}$

$\small\bold{\underline{\sf{\green{To\:Prove:-}}}}$

$\\ \implies { \bold{ \dfrac{ \cos(a) - \sin(a) + 1}{ \cos(a) + \sin(a) - 1} = cosec(a) + \cot(a) }}$

$\small\bold{\underline{\sf{\green{Solution:-}}}}$

L.H.S

$\\ \: \: = \: \: { \bold{ \dfrac{ \cos(a) - \sin(a) + 1}{ \cos(a) + \sin(a) - 1} }}$

( Rationalization of denominator)

$\\ \: \: = \: \: { \bold{ \dfrac{ \cos(a) - \sin(a) + 1}{ \cos(a) + \sin(a) - 1} \times \dfrac{ \cos(a) + \sin(a) + 1}{\cos(a) + \sin(a) + 1} }}$

$\\ \: \: = \: \: { \bold{ \dfrac{ \{ \cos(a) - \sin(a) + 1 \} \{\cos(a) + \sin(a) + 1 \}}{ \{\cos(a) + \sin(a) \}^{2} - {(1)}^{2} }}}$

$\\ \: \: = \: \: { \bold{ \dfrac{ \cos^{2} (a) + \sin(a) \cos(a) + \cos(a) - \sin(a) \cos(a) - \sin^{2} (a) - \sin(a) + \cos(a) + \sin(a) + 1}{ \cos^{2} (a) + \sin^{2} (a) + 2 \sin(a) \cos(a) - 1}}}$

$\\ \: \: = \: \: { \bold{ \dfrac{ \cos^{2} (a) + 2 \cos(a) - \sin^{2} (a) + 1}{ 1 + 2 \sin(a) \cos(a) - 1}}}$

$\\ \: \: = \: \: { \bold{ \dfrac{ \cos^{2} (a) + 2 \cos(a) - \sin^{2} (a) + 1}{2 \sin(a) \cos(a) }}}$

$\\ \: \: = \: \: { \bold{ \dfrac{ \cos^{2} (a) + 2 \cos(a) - (1 - \cos^{2} (a)) + 1}{2 \sin(a) \cos(a) }}}$

$\\ \: \: = \: \: { \bold{ \dfrac{ \cos^{2} (a) + 2 \cos(a) - 1 + \cos^{2} (a) + 1}{2 \sin(a) \cos(a) }}}$

$\\ \: \: = \: \: { \bold{ \dfrac{2 \cos^{2} (a) + 2 \cos(a)}{2 \sin(a) \cos(a) }}}$

$\\ \: \: = \: \: { \bold{ \dfrac{ \cos (a) + 1}{ \sin(a) }}}$

$\\ \: \: = \: \: { \bold{ \dfrac{1}{ \sin(a) } + \dfrac{ \cos (a) }{ \sin(a) }}}$

$\\ \: \: = \: \: { \bold{cosec(a) + \cot(a) }}$

= R.H.S

Hence,

L.H.S = R.H.S

( Proved)