Rationalize the denominator and simplify: cos^2x / 1 - sinx

Question

This is for my trig class. Thanks for the help!

Answer 1

cos^2(x) / (1 - sin(x)) =

(1 - sin^2(x)) / (1 - sin(x)) =

(1 + sin(x)) * (1 - sin(x)) / (1 - sin(x)) =

1 + sin(x), x ≠ pi/2 + 2pi(n), n integer.

Answer 2

cos^2(x)/[1-sin(x)]

multiply numerator and denominator by 1+sin(x)

[cos^2(x)][1+sin(x)]/[1-sin(x)][1+sin(x)]

cos^2(x)+cos^2(x)sin(x)/[1-sin^2(x)]

cos^2(x)+cos^2(x)sin(x)/cos^2(x) because 1=sin^2(x)+cos^2(x)

1+sin(x) is the answer

or

cos^2(x)/[1-sin(x)]=[1-sin^2(x)]/[1-sin(x)]

                             =[1-sin(x)][1+sin(x)]/[1-sin(x)]

                             =1+sin(x)

Answer 3

 

  COS ^2 x / (1 -SIN X )  =

 

  Cos ^2 X . ( 1 + SIN X)  /.  (1 + Sin X) (1 - SIN X) =

  

                                    /=  1- Sin^2 X

                                    / =  Cos ^2X

          ! + Sin X.

                              

 

  Rationalization is not a proper word to use here. Because ( 1- Sin X) is not an irrational quantity.