MATH SOLVE

4 months ago

Q:
# URGENT!!! Which expression is equivalent to cos(2α)cosα−sinα for all values of α for which cos(2α)cosα−sinα is defined?Select the correct answer below:2cosα+sinαcot2α−2cos2α2tanα1+tanα2sinα1−tan2α

Accepted Solution

A:

ANSWER[tex] \cos( \alpha ) + \sin( \alpha ) [/tex]EXPLANATIONThe 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]