Using Complex exponential definitions of sine and cosine, prove cos ⁡ 2 θ = cos 2 ⁡ θ − sin 2 ⁡ θ

Question

Using Complex exponential definitions of sine and cosine, prove   cos  ⁡  2  θ  =      cos    2    ⁡  θ  −      sin    2    ⁡  θ

crociandomh · Accepted Answer

Hint:  cos  ⁡  2  θ  +  i  sin  ⁡  2  θ  =      e          2      i      θ        =            (              e                  i          θ                    )        2    =  (  cos  ⁡  θ  +  i  sin  ⁡  θ      )    2  Then expand the right side and compare real parts

Eden Solomon · Accepted Answer

cos        ⁡        (        2        θ        )        +        i        sin        ⁡        (        2        θ        )                    =                              e                      2            i            θ                                                    =                    (                  e                      i            θ                                    )          2                                          =                    (        cos        ⁡        θ        +        i        sin        ⁡        θ                  )          2                                          =                    (        cos        ⁡        θ                  )          2                +        2        i        cos        ⁡                  θ                sin        ⁡                  θ                +                  i          2                (        sin        ⁡                  θ                          )          2                                          =                    (        cos        ⁡                  θ                          )          2                +        2        i        cos        ⁡                  θ                sin        ⁡                  θ                −        (        sin        ⁡                  θ                          )          2                    Equating real and imaginary parts gives us:                    cos        ⁡        (        2                  θ                )        =        (        cos        ⁡                  θ                          )          2                −        (        sin        ⁡                  θ                          )          2                                    sin        ⁡        (        2                  θ                )        =        2        cos        ⁡                  θ                sin        ⁡                  θ

Using Complex exponential definitions of sine and cosine, prove cos ⁡ 2

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