Prove that sin 6 ⁡ θ 2 + cos 6 ⁡ θ 2 = 1 4 ( 1 + 3 cos 2 ⁡ θ )

Question

Prove that       sin          6        ⁡            θ      2        +      cos          6        ⁡            θ      2        =      1    4    (  1  +  3      cos    2    ⁡  θ  )

grcalia1 · Accepted Answer

sin    6    ⁡      θ    2    +      cos    6    ⁡      θ    2    =  ⋯  =  1  −  3      (    1    −          cos      2        ⁡          θ      2        )        cos    2    ⁡      θ    2  is correct. From here, note that                              sin          6                ⁡                  θ          2                +                  cos          6                ⁡                  θ          2                                    =        1        −        3                  (          1          −                                    cos              2                        ⁡                          θ              2                                )                                      cos            2                    ⁡                      θ            2                                                            =        1        −        3                  (          1          −                                                    cos                ⁡                θ                +                1                            2                                )                ⋅                                            cos              ⁡              θ              +              1                        2                                                            =        1        −        3        ⋅                              1            −            cos            ⁡            θ                    2                ⋅                              1            +            cos            ⁡            θ                    2                                                  =                              4            −            3            (            1            −                          cos              2                        ⁡            θ            )                    4                                                  =                              1            +            3                          cos              2                        ⁡            θ                    4

Zion Wheeler · Accepted Answer

You may begin factoring:                              sin          6                ⁡                  θ          2                +                  cos          6                ⁡                  θ          2                                    =                              (                                    sin          2                ⁡                  θ          2                +                  cos          2                ⁡                  θ          2                                      )                                                (                                    sin          4                ⁡                  θ          2                −                  sin          2                ⁡                  θ          2                          cos          2                ⁡                  θ          2                +                  cos          4                ⁡                  θ          2                                      )                                                            =                              (                                    cos          2                ⁡                  θ          2                −                  sin          2                ⁡                  θ          2                                                    )                                2                +                  sin          2                ⁡                  θ          2                          cos          2                ⁡                  θ          2                =                  cos          2                ⁡        θ        +                  1          4                          sin          2                ⁡        θ                                          =                  1          4                (        3                  cos          2                ⁡        θ        +        1        )        .            You also may linearise:      cos    2    ⁡  θ  +      1    4        sin    2    ⁡  θ  =      1    2    (  1  +  cos  ⁡  2  θ  )  +      1    8    (  1  −  cos  ⁡  2  θ  )  =      1    8    (  3  cos  ⁡  2  θ  +  5  )  .

Prove that sin <mrow class="MJX-TeXAtom-ORD"> 6 </mrow> </msup> &#x

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