Calculate trigonometry expression: cos ⁡ 3 α − sin ⁡ 3 α cos ⁡ α + sin ⁡ α if cos ⁡ 3 α − sin ⁡ 3 α cos ⁡ α + sin ⁡ α , if sin ⁡ ( π 4 − α ) = 0 , 1.I make following transformation: cos ⁡ 3 α − sin ⁡ 3 α cos ⁡ α + sin ⁡ α = 4 cos 3 ⁡ α − 3 cos ⁡ α − 3 sin ⁡ α + 4 sin 3 ⁡ α cos ⁡ α + sin ⁡ α = = 4 ( cos 3 ⁡ α + sin 3 ⁡ α ) − 3 ( cos ⁡ α + sin ⁡ α ) cos ⁡ α + sin ⁡ α = = 4 ( cos ⁡ α + sin ⁡ α ) ( cos 2 ⁡ α − cos ⁡ α ⋅ sin ⁡ α + sin 2 ⁡ α ) − 3 ( cos ⁡ α + sin ⁡ α ) cos ⁡ α + sin ⁡ α = ( cos ⁡ α + sin ⁡ α ) [ 4 ( 1 − sin ⁡ α cos ⁡ α ) − 3 ] cos ⁡ α + sin ⁡ α = 4 − 4 sin ⁡ α cos ⁡ α − 3 = 1 − 2 sin ⁡ 2 α . What next? How i can use substitution?

Question

Calculate trigonometry expression:             cos      ⁡      3      α      −      sin      ⁡      3      α              cos      ⁡      α      +      sin      ⁡      α       if                      cos        ⁡        3        α        −        sin        ⁡        3        α                    cos        ⁡        α        +        sin        ⁡        α              ,            if          sin  ⁡      (                  π        4              −    α    )    =  0  ,  1.I make following transformation:                                                        cos              ⁡              3              α              −              sin              ⁡              3              α                                      cos              ⁡              α              +              sin              ⁡              α                                      =                                            4                              cos                3                            ⁡              α              −              3              cos              ⁡              α              −              3              sin              ⁡              α              +              4                              sin                3                            ⁡              α                                      cos              ⁡              α              +              sin              ⁡              α                                      =                            =                                            4                              (                                  cos                  3                                ⁡                α                +                                  sin                  3                                ⁡                α                )                            −              3                              (                cos                ⁡                α                +                sin                ⁡                α                )                                                    cos              ⁡              α              +              sin              ⁡              α                                      =                            =                                            4                              (                cos                ⁡                α                +                sin                ⁡                α                )                                            (                                  cos                  2                                ⁡                α                −                cos                ⁡                α                ⋅                sin                ⁡                α                +                                  sin                  2                                ⁡                α                )                            −              3                              (                cos                ⁡                α                +                sin                ⁡                α                )                                                    cos              ⁡              α              +              sin              ⁡              α                                      =                                                                                (                cos                ⁡                α                +                sin                ⁡                α                )                                            [                                4                                (                1                −                sin                ⁡                α                cos                ⁡                α                )                −                3                                ]                                                    cos              ⁡              α              +              sin              ⁡              α                                      =        4        −        4        sin        ⁡        α        cos        ⁡        α        −        3        =        1        −        2        sin        ⁡        2        α        .            What next? How i can use substitution?

Giovanna Erickson · Accepted Answer

Notice that  1  −  2  sin  ⁡  2  α  =  1  −  2  cos  ⁡      (    2          (              π        4            −      α      )        )    =  1  −  2      1    −          sin      2        ⁡          (      2              (                  π          4                −        α        )            )

uplakanimkk · Accepted Answer

Notice,If  sin  ⁡      (          π      4        −    α    )    =  0  sin  ⁡  α  cos  ⁡      π    4    −  cos  ⁡  α  sin  ⁡      π    4    =  0    ⟹    tan  ⁡  α  =  1Hence,      1    −    2    sin    ⁡    2    α    =  1  −  2            2      tan      ⁡      α              1      +              tan        2            ⁡      α        =  1  −  2            2      (      1      )              1      +      (      1              )        2              =      −    1  If  sin  ⁡      (          π      4        −    α    )    =  1    ⟹    α  =  −      (    2    n    π    +          π      4        )  Hence,      1    −    2    sin    ⁡    2    α    =  1  −  2  sin  ⁡  2      (    −          (      2      n      π      +              π        4            )        )    =  1  +  2  sin  ⁡      (    4    n    π    +          π      2        )    =  1  +  2  sin  ⁡      π    2    =  1  +  2  (  1  )  =      3

Calculate trigonometry expression: cos ⁡ 3 &#x03B1

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