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

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3 Answers

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.
 
  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.
 
 

Comments

A rational number can be expressed as a fraction of integers.
 
Can you do that with 1 - sin(x) for all x?
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)