How would you show cos 2 ⁡ ( x ) sin 3 ⁡ ( x ) = 1 16 ( 2 sin ⁡ ( x ) + sin ⁡ ( 3 x ) − sin ⁡ ( 5 x ) )

Question

How would you show       cos    2    ⁡  (  x  )      sin    3    ⁡  (  x  )  =      1    16    (  2  sin  ⁡  (  x  )  +  sin  ⁡  (  3  x  )  −  sin  ⁡  (  5  x  )  )

Lilliana Burton · Accepted Answer

cos    2    ⁡  (  x  )      sin    3    ⁡  (  x  )  is a   2  π-periodic odd function with a finite expansion as a Fourier sine series.By exploiting De Moivre&#039;s identities  sin  ⁡  (  x  )  =                    e                  i          x                    −              e                  −          i          x                            2      i        ,    cos  ⁡  (  x  )  =                    e                  i          x                    +              e                  −          i          x                      2  and the binomial theorem, the claim follows by straightforward algebra: it is essentially the same as computing the coefficients of the polynomial   (      t    2    +  1      )    2    (      t    2    −  1      )    3

Leland Morrow · Accepted Answer

F  =      cos    2    ⁡  x      sin    3    ⁡  x  =            sin      ⁡      x        4    (  2  sin  ⁡  x  cos  ⁡  x      )    2  as   sin  ⁡  2  x  =  2  sin  ⁡  x  cos  ⁡  x  F  =                    sin        ⁡        x        (        1        −        cos        ⁡        4        x        )            8      as   cos  ⁡  2  A  =  1  −  2      sin    2    ⁡  ANow,  sin  ⁡  x  (  1  −  cos  ⁡  4  x  )  =                    2        sin        ⁡        x        −        2        sin        ⁡        x        cos        ⁡        4        x            2        =                    2        sin        ⁡        x        −        (        sin        ⁡        5        x        −        sin        ⁡        3        x        )            2      using Werner Formulas

How would you show cos 2 </msup> ⁡ ( x ) sin

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