Is there a formula that relates cos ⁡ ( p q x ) , p ∈ Z and q ∈ N , to cos ⁡ x?I thought of cos 2 ⁡ ( p q x ) = 1 + cos ⁡ ( 2 p q x ) 2 and it doesn't leave here, only and only in case of p = 1 and q = 2

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Is there a formula that relates   cos  ⁡      (          p      q        x    )  ,   p  ∈      Z   and   q  ∈      N  , to   cos  ⁡  x?I thought of       cos    2    ⁡      (          p      q        x    )    =            1      +      cos      ⁡              (                              2            p                    q                x        )              2   and it doesn&#039;t leave here, only and only in case of   p  =  1 and   q  =  2

Kennedy Evans · Accepted Answer

Let   cos  ⁡  (  x  )  =  c, so   sin  ⁡  (  x  )  =  ±      1    −          c      2       and   exp  ⁡  (  ±  i  x  )  =  c  ±  i      1    −          c      2       (not necessarily with the same + and −). Then   exp  ⁡  (  ±  i  p  x      /    q  )  =  (  c  ±  i      1    −          c      2            )          p              /            q       so  cos  ⁡  (  p  x      /    q  )  =            exp      ⁡      (      i      p      x              /            q      )      +      exp      ⁡      (      −      i      p      x              /            q      )        2    =            (      c      +      i              1        −                  c          2                            )                  p                      /                    q                    +      (      c      −      i              1        −                  c          2                            )                  p                      /                    q                      2  Of course the multi-valued nature of the p/q power of complex numbers must be taken into account.

Is there a formula that relates cos(p/q x), p in Z and q in N, to cos x ?

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