Theorem: Product Formula for Sine

:$\ds \map \sin {n z} = 2^{n - 1} \prod_{k \mathop = 0}^{n - 1} \map \sin {z + \frac {k \pi} n}$

Proof:
From Gauss Multiplication Formula, we have:
:$\ds \forall z \notin \set {-\frac m n: m \in \N}: \prod_{k \mathop = 0}^{n - 1} \map \Gamma {z + \frac k n} = \paren {2 \pi}^{\paren {n - 1} / 2} n^{1/2 - n z} \map \Gamma {n z}$
Therefore:
:$\ds \map \Gamma {n z} = \paren {2 \pi}^{\paren {1 - n} / 2} n^{n z - 1/2} \prod_{k \mathop = 0}^{n - 1} \map \Gamma {z + \frac k n} $
From Euler's Reflection Formula, we have:
:$\forall z \notin \Z: \map \Gamma z \map \Gamma {1 - z} = \dfrac \pi {\map \sin {\pi z} }$
Therefore:
{{begin-eqn}}
{{eqn | l = \frac {\map \sin {\pi z} } \pi
      | r = \frac 1 {\map \Gamma z \map \Gamma {1 - z} }
      | c = Euler's Reflection Formula
}}
{{eqn | ll = \leadsto
      | l = \frac {\map \sin {n z} } \pi
      | r = \frac 1 {\map \Gamma {\dfrac {n z} \pi } \map \Gamma {1 - \dfrac {n z} \pi} }
      | c = $z \to \dfrac {n z} \pi$
}}
{{eqn | r = \frac 1 {\map \Gamma {\dfrac {n z} \pi } \paren {-\dfrac {n z} \pi}  \map \Gamma {- \dfrac {n z} \pi} }
      | c = Gamma Difference Equation
}}
{{eqn | ll = \leadsto
      | l = \map \sin {n z} 
      | r = \frac {- \pi^2} {\map \Gamma {\dfrac {n z} \pi } \paren {n z}  \map \Gamma {- \dfrac {n z} \pi} }
      | c = multiplying both sides by $\pi$
}}
{{eqn | r = \dfrac {- \pi^2} {\paren {\paren {2 \pi}^{\paren {1 - n} / 2} n^{n \paren {z / \pi} - 1/2} \ds \prod_{k \mathop = 0}^{n - 1} \map \Gamma {\dfrac z \pi + \dfrac k n} } \paren {n z}  \paren {\paren {2 \pi}^{\paren {1 - n} / 2} n^{n \paren {-z / \pi} - 1/2} \ds \prod_{k \mathop = 0}^{n - 1} \map \Gamma {\dfrac {-z} \pi + \dfrac k n} } }
      | c = Gauss Multiplication Formula
}}
{{eqn | r = \dfrac {- \pi^2 \paren {2 \pi}^{\paren {n - 1} } } {\paren {n^{n \paren {z / \pi} - 1/2} \ds \prod_{k \mathop = 0}^{n - 1} \map \Gamma {\dfrac z \pi + \dfrac k n} } \paren {n z}  \paren {n^{n \paren {-z / \pi} - 1/2} \ds \prod_{k \mathop = 0}^{n - 1} \map \Gamma {\dfrac {-z} \pi + \dfrac k n} } }
      | c = moving $2 \pi$ to the top and Product of Powers
}}
{{eqn | r = \dfrac {- \pi^2 \paren {2 \pi}^{\paren {n - 1} } } {\paren {\ds \prod_{k \mathop = 0}^{n - 1} \map \Gamma {\dfrac z \pi + \dfrac k n} } \paren z  \paren {\ds \prod_{k \mathop = 0}^{n - 1} \map \Gamma {\dfrac {-z} \pi + \dfrac k n} } }
      | c = $n$ cancels on the bottom and Product of Powers
}}
{{eqn | r = 2^{n - 1} \dfrac {- \pi^{\paren {n + 1} } } {z \ds \prod_{k \mathop = 0}^{n - 1} \map \Gamma {\dfrac z \pi + \dfrac k n} \map \Gamma {\dfrac {-z} \pi + \dfrac k n} }
      | c = 
}}
{{eqn | r = 2^{n - 1} \dfrac {\pi^n} {\dfrac {-z} \pi \map \Gamma {\dfrac z \pi} \map \Gamma {\dfrac {-z} \pi} \ds \prod_{k \mathop = 1}^{n - 1} \map \Gamma {\dfrac z \pi + \dfrac k n} \map \Gamma {\dfrac {-z} \pi + \dfrac k n} }
      | c = borrowing a $-\pi$ from the top
}}
{{eqn | r = 2^{n - 1} \dfrac {\pi^n} {\map \Gamma {\dfrac z \pi} \map \Gamma {\dfrac {1 - z} \pi} \ds \prod_{k \mathop = 1}^{n - 1} \map \Gamma {\dfrac z \pi + \dfrac k n} \map \Gamma {\dfrac {-z} \pi + \dfrac k n} }
      | c = Gamma Difference Equation
}}
{{eqn | r = 2^{n - 1} \dfrac {\pi^n} {\dfrac {\pi} {\map \sin {\pi \paren {\dfrac z \pi } } } \ds \prod_{k \mathop = 1}^{n - 1} \map \Gamma {\dfrac z \pi + \dfrac k n} \map \Gamma {\dfrac {-z} \pi + \dfrac k n} }
      | c = Euler's Reflection Formula
}}
{{eqn | r = 2^{n - 1} \dfrac {\pi^n} {\dfrac {\pi} {\map \sin {\pi \paren {\dfrac z \pi } } } \ds \prod_{k \mathop = 1}^{n - 1} \map \Gamma {\dfrac z \pi + \dfrac k n} \map \Gamma {\dfrac {-z} \pi + \dfrac k n + 1 - 1} }
      | c = adding $0$
}}
{{eqn | r = 2^{n - 1} \dfrac {\pi^n} {\dfrac {\pi} {\map \sin {\pi \paren {\dfrac z \pi } } } \ds \prod_{k \mathop = 1}^{n - 1} \map \Gamma {\dfrac z \pi + \dfrac k n} \map \Gamma {1 - \paren {\dfrac z \pi + \dfrac {n - k} n} } }
      | c = 
}}
{{eqn | r = 2^{n - 1} \dfrac {\pi^n } {\ds \prod_{k \mathop = 0}^{n - 1} \frac {\pi} {\map \sin {\pi \paren {\dfrac z \pi + \dfrac k n} } }  }
      | c = Euler's Reflection Formula
}}
{{eqn | r = 2^{n - 1} \dfrac {\pi^n } {\pi^n \ds \prod_{k \mathop = 0}^{n - 1} \frac 1 {\map \sin {z + \dfrac {k \pi} n} }  }
      | c = 
}}
{{eqn | r = 2^{n - 1} \prod_{k \mathop = 0}^{n - 1} \map \sin {z + \dfrac {k \pi} n}
      | c = 
}}
{{eqn | ll = \leadsto
      | l = \map \sin {n z}
      | r = 2^{n - 1} \prod_{k \mathop = 0}^{n - 1} \map \sin {z + \dfrac {k \pi} n}
      | c = 
}}
{{end-eqn}}
{{qed}}