Rationalizing 1 + cos ⁡ x + 1 − cos ⁡ x 1 + cos ⁡ x − 1 − cos ⁡ x in two ways gives different answerswe get answer 1 + sin ⁡ x cos ⁡ x but when we rationalize numerator we get cos ⁡ x 1 + sin ⁡ x How is this possible, because rationalizing means just multiplying by 1?

Question

Rationalizing                     1        +        cos        ⁡        x            +              1        −        cos        ⁡        x                            1        +        cos        ⁡        x            −              1        −        cos        ⁡        x             in two ways gives different answerswe get answer            1      +      sin      ⁡      x              cos      ⁡      x      but when we rationalize numerator we get            cos      ⁡      x              1      +      sin      ⁡      x      How is this possible, because rationalizing means just multiplying by 1?

indyterpep · Accepted Answer

You might have an error, check that at the end you should have that rationalazing the denominator you should have that                                                  1          +          sin          ⁡                      x                                                                            cos          ⁡                      x                                ,and the numerator you should have that                                                  cos          ⁡                      x                                                                            1          −          sin          ⁡                      x                                ,wich it´s always the same by   (  cos  ⁡      x        )    2    +  (  sin  ⁡      x        )    2    =  1

Jaelyn Payne · Accepted Answer

Putting a for cosx, you have                              1          +          a                +                  1          −          a                                      1          +          a                −                  1          −          a                    I now do the rationalizing (using   (  u  +  v  )  (  u  −  v  )  =      u    2    −      v    2    ) and later substitute   a  =  cos  ⁡  (  x  ) and use       sin    2    ⁡  (  x  )  +      cos    2    ⁡  (  x  )  =  1I am doing this in excruciating detail so you can see all the steps involved. Once you understand these, you should be able to do this kind of thing by yourself.                                                                        1                +                a                            +                              1                −                a                                                                    1                +                a                            −                              1                −                a                                                                =                                                            1                +                a                            +                              1                −                a                                                                    1                +                a                            −                              1                −                a                                                                                                        1                +                a                            +                              1                −                a                                                                    1                +                a                            +                              1                −                a                                                                              =                                            1              +              a              +              2                              1                +                a                                            1                −                a                            +              1              −              a                                      (              1              +              a              )              −              (              1              −              a              )                                                                =                                            2              +              2                              (                1                +                a                )                (                1                −                a                )                                                    2              a                                                                =                                            1              +                              1                −                                  a                  2                                                      a                                                    =                                            1              +                              1                −                                  cos                  2                                ⁡                (                x                )                                                    cos              ⁡              (              x              )                                                                =                                            1              +              sin              ⁡              (              x              )                                      cos              ⁡              (              x              )

Rationalizing (sqrt(1+cos x)+sqrt(1−cos x))/(sqrt(1+cos x)−sqrt(1−cos x)) in two ways gives different answers

Answered question

Answer & Explanation

New Questions in Trigonometry