Question

Prove that cos A – sin A + 1 / cos A + sin A – 1 = cosec A + cot A using the identity cosec² A = 1 + cot² A. Please answer quickley

Answer 1

Answer:

Hope this helps and sorry for the bad handwriting I was in a hurry

Your Welcome

Answer 2

Step-by-step explanation:

$\green{question::}$

${prove\:that\::}$

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

$since \: we \: need \: the \: answer \: in \: \\ cosec \: and \: cot \: we \: will \: divide \: it \\ by \: sin$

$thus \: we \: have$

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

$\frac{ \cot(a) - 1 + \csc(a) }{ \cot(a) + 1 - \csc(a) }$

$we \: can \: replace \: 1 \: as \: { \csc(a) }^{2} - { \cot(a) }^{2}$

$\frac{ \cot(a) + \csc(a) - ( {\csc }^{2}a - { \cot(a) }^{2} ) }{ \cot(a) + 1 - \csc(a) }$

taking common, we have

$\frac{ ( \cot(a) + \csc(a) )(1 - \csc(a) + \cot(a)}{ 1 - \csc(a) + \cot(a) }$

$\csc(a) + \cot(a)$

$\red{hence\:proved}$