If sin 2 ⁡ x + sin 2 ⁡ 2 x + sin 2 ⁡ 3 x = 1, what does cos 2 ⁡ x + cos 2 ⁡ 2 x + cos 2 ⁡ 3 x equal?

Question

If       sin    2    ⁡  x +       sin    2    ⁡  2  x +       sin    2    ⁡  3  x = 1, what does       cos    2    ⁡  x +       cos    2    ⁡  2  x +       cos    2    ⁡  3  x equal?

Darrell Valencia · Accepted Answer

You know that            sin        2    x  +            sin        2    2  x  +            sin        2    3  x  =  1    (      Eq.    1  )Notice that                                          sin                    2                x        +                              cos                    2                x        =        1        ⇒                              sin                    2                x        =        1        −                              cos                    2                x            Substituting the identity in Equation (1):                                                                                                              sin                                    2                                x                            ⏟                                            1            −                                          cos                            2                        x                                              +                                                                                                  sin                                    2                                2                x                            ⏟                                            1            −                                          cos                            2                        2            x                          +                                                                                                  sin                                    2                                3                x                            ⏟                                            1            −                                          cos                            2                        3            x                          =        1                                                        ⇒        1        −                              cos                    2                x        +        1        −                              cos                    2                2        x        +        1        −                              cos                    2                3        x        =        1                                          ⇒        3        −                  (                                    cos                        2                    x          +                                    cos                        2                    2          x          +                                    cos                        2                    3          x          )                =        1                                          ⇒        2        =                              cos                    2                x        +                              cos                    2                2        x        +                              cos                    2                3        x            Therefore:            cos        2    x  +            cos        2    2  x  +            cos        2    3  x  =  2

Yesenia Obrien · Accepted Answer

sin    2    ⁡  x  +      sin    2    ⁡  2  x  +      sin    2    ⁡  3  x  =  1    ⟹    (  1  −  c  o      s    2    x  )  +  (  1  −      cos    2    ⁡  2  x  )  +  (  1  −  c  o      s    2    3  x  )  =  1 Rearranging terms we get       cos    2    ⁡  x  +      cos    2    ⁡  2  x  +      cos    2    ⁡  3  x  =  2

If sin 2 </msup> ⁡ x + sin 2 </msup>

Answered question

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