Question

cos a - sin a +1 / cos a +.sin a -1 =cosec a +cot a. prove it​

Answer 1

TO PROVE :–

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

SOLUTION :–

• Let's take L.H.S. –

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

• Rationalization of denominator –

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

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

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

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

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

• We should write this as –

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

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

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

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

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

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

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

$\\ \: \: = \: \: { \bold{R.H.S.}}$

$\\ \: \: \: \: \: \: { \underbrace{ \bold{Hence \: \: proved}}}$

Answer 2

Step-by-step explanation:

${\red{\underline{\underline{\bold{To\:prove:-}}}}}$

$\frac{ \cos(a) - \sin(a) + 1 }{ \cos(a) + \sin(a) - 1 }= cosec(a) - cot(a)$

${\blue{\underline{\underline{\bold{Proof:-}}}}}$

Let us verify by simplifying L.H.S and R.H.S separately.

$L.H.S = \frac{ \cos(a) - \sin(a) + 1 }{ \cos(a) + \sin(a) - 1}$

Divide throughout by sin(a):-

$= \frac{ \frac{ \cos(a) }{ \sin(a) } - \frac{ \sin(a) }{ \sin(a) } + \frac{1}{ \sin(a) } }{ \frac{ \cos(a) }{ \sin(a) } + \frac{ \sin(a) }{ \sin(a) } - \frac{1}{ \sin( a) } } \\\\ = \frac{ \cot(a) - 1 + \cosec(a) }{ \cot(a) - 1 + \cosec(a) } \\\\ = \frac{ [\cot( a) - (1 - \cosec(a) )][(\cot( a) - (1 - \cosec(a) )] }{ [\cot(a ) + (1 - \cosec(a))][( \cot( a) - (1 - \cosec(a) )]} \\\\</p><p></p><p>= \frac{ {[ \cot(a) - 1 + \cosec(a) ] }^{2} }{ { [\cot(a)] }^{2} - {[1 - \cosec(a)] }^{2} } \\\\ = \frac{ \cot(a) + 1 + {cosec}^{2}(a) - 2 \cot(a) - 2 \cosec(a) + 2 \cot( a) \cosec(a) }{ {cot}^{2}a - (1 + {cosec}^{2} a + 2 \cosec(a) )} \\\\ = \frac{2 {cosec}^{2}a + 2 \cot(a) \cosec(a) - 2 \cot(a) - 2 \cosec(a) }{ {cot}^{2} a - 1 - {cosec}^{2}a + 2 \cosec(a) } \\\\ = \frac{2 \cosec(a)( \cosec(a) + \cot(a) ) - 2 \cot(a) + \cosec(a) }{ {cot}^{2} a - {cosec}^{2}a - 1 +2 \cosec(a) }\: \: \: \: ( \therefore {cot}^{2}a - {cosec}^{2} a \: = - 1) \\\\ = \frac{ (\cosec(a) + \cot(a))(2 \cosec(a) - 2 ) }{ - 1 - 1 + 2 cosec(a) } \\\\ = \frac{ \cosec(a) + \cot(a) (2 cosec(a) - 2)}{(2cosec(a) - 2)} \\\\ = cosec(a) + cot(a)$

= R.H.S

Hence proved