ANSWER
[tex] \cos( \alpha ) + \sin( \alpha ) [/tex]
EXPLANATION
The given expression is
[tex] \frac{ \cos(2 \alpha ) }{ \cos( \alpha ) - \sin( \alpha ) } [/tex]
Recall and use the double angle identity,
[tex] \cos(2 \alpha ) = { \cos}^{2} \alpha - { \sin}^{2} \alpha [/tex]
This implies that,
[tex]\frac{ \cos(2 \alpha ) }{ \cos( \alpha ) - \sin( \alpha ) } = \frac{ \ { \cos}^{2} \alpha - { \sin}^{2} \alpha }{ \cos( \alpha ) - \sin( \alpha ) } [/tex]
Recall again that: a²-b²=(a-b)(a+b)
We use difference of two squares to obtain,
[tex]\frac{ \cos(2 \alpha ) }{ \cos( \alpha ) - \sin( \alpha ) } = \frac{ (\ { \cos}\alpha - { \sin}\alpha )(\ { \cos}\alpha + { \sin}\alpha)}{ \cos( \alpha ) - \sin( \alpha ) } [/tex]
We cancel out the common factors to get,
[tex]\frac{ \cos(2 \alpha ) }{ \cos( \alpha ) - \sin( \alpha ) } = { \cos}\alpha + { \sin}\alpha[/tex]
