Theorem: Sum over Integers of Sine of n + alpha of theta over n + alpha

Let $\alpha \in \R$ be a real number which is specifically not an integer.
For $0 < \theta < 2\pi$:
:$\ds \sum_{n \mathop \in \Z} \dfrac {\map \sin {n + \alpha} \theta} {n + \alpha} = \pi$

Proof:
First we establish the following, as they will be needed later.
{{begin-eqn}}
{{eqn | o = 
      | r = \map \sin {\alpha + n} \theta + \map \sin {\alpha - n} \theta
      | c = 
}}
{{eqn | r = 2 \map \sin {\dfrac {\paren {\alpha + n} \theta + \paren {\alpha - n} \theta} 2} \map \cos {\dfrac {\paren {\alpha + n} \theta - \paren {\alpha - n} \theta} 2}
      | c = Sine plus Sine
}}
{{eqn | n = 1
      | r = 2 \sin \alpha \theta \cos n \theta
      | c = simplification
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | o = 
      | r = \map \sin {\alpha + n} \theta - \map \sin {\alpha - n} \theta
      | c = 
}}
{{eqn | r = 2 \map \cos {\dfrac {\paren {\alpha + n} \theta + \paren {\alpha - n} \theta} 2} \map \sin {\dfrac {\paren {\alpha + n} \theta - \paren {\alpha - n} \theta} 2}
      | c = Sine minus Sine
}}
{{eqn | n = 2
      | r = 2 \cos \alpha \theta \sin n \theta
      | c = simplification
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | o = 
      | r = \sum_{n \mathop = -\infty}^\infty \dfrac {\map \sin {n + \alpha} \theta} {n + \alpha}
      | c = 
}}
{{eqn | r = \dfrac {\map \sin {0 + \alpha} \theta} {0 + \alpha} + \sum_{n \mathop = 1}^\infty \dfrac {\map \sin {n + \alpha} \theta} {n + \alpha} + \sum_{n \mathop = -\infty}^{-1} \dfrac {\map \sin {n + \alpha} \theta} {n + \alpha}
      | c = 
}}
{{eqn | r = \dfrac {\sin \alpha \theta} \alpha + \sum_{n \mathop = 1}^\infty \dfrac {\map \sin {n + \alpha} \theta} {n + \alpha} + \sum_{n \mathop = 1}^\infty \dfrac {\map \sin {-n + \alpha} \theta} {-n + \alpha}
      | c = 
}}
{{eqn | r = \dfrac {\sin \alpha \theta} \alpha + \sum_{n \mathop = 1}^\infty \paren {\dfrac {\map \sin {\alpha + n} \theta} {\alpha + n} + \dfrac {\map \sin {\alpha - n} \theta} {\alpha - n} }
      | c = 
}}
{{eqn | r = \dfrac {\sin \alpha \theta} \alpha + \sum_{n \mathop = 1}^\infty \paren {\dfrac {\paren {\alpha - n} \map \sin {\alpha + n} \theta + \paren {\alpha + n} \map \sin {\alpha - n} \theta} {\alpha^2 - n^2} }
      | c = 
}}
{{eqn | r = \dfrac {\sin \alpha \theta} \alpha + \sum_{n \mathop = 1}^\infty \paren {\dfrac {\alpha \map \sin {\alpha + n} \theta - n \map \sin {\alpha + n} \theta + \alpha \map \sin {\alpha - n} \theta + n \map \sin {\alpha - n} \theta} {\alpha^2 - n^2} }
      | c = 
}}
{{eqn | r = \dfrac {\sin \alpha \theta} \alpha + \sum_{n \mathop = 1}^\infty\paren {\dfrac {\alpha \paren {\map \sin {\alpha + n} \theta + \map \sin {\alpha - n} \theta} - n \paren {\map \sin {\alpha + n} \theta - \map \sin {\alpha - n} \theta} } {\alpha^2 - n^2} }
      | c = 
}}
{{eqn | r = \dfrac {\sin \alpha \theta} \alpha + \sum_{n \mathop = 1}^\infty \paren {\dfrac {2 \alpha \sin \alpha \theta \cos n \theta - 2 n \cos \alpha \theta \sin n \theta} {\alpha^2 - n^2} }
      | c = from $(1)$ and $(2)$
}}
{{eqn | r = \dfrac {\sin \alpha \theta} \alpha + 2 \sin \alpha \theta \sum_{n \mathop = 1}^\infty \dfrac {\alpha \cos n \theta } {\alpha^2 - n^2} - 2 \cos \alpha \theta \sum_{n \mathop = 1}^\infty \dfrac {n \sin n \theta} {\alpha^2 - n^2}
      | c = Linear Combination of Indexed Summations
}}
{{end-eqn}}
Setting $\theta = \pi$:
{{begin-eqn}}
{{eqn | l = \sum_{n \mathop = -\infty}^\infty \dfrac {\map \sin {n + \alpha} \pi} {n + \alpha}
      | r = \dfrac {\sin \alpha \pi} \alpha + 2 \sin \alpha \pi \sum_{n \mathop = 1}^\infty \dfrac {\alpha \cos n \pi } {\alpha^2 - n^2} - 2 \cos \alpha \pi \sum_{n \mathop = 1}^\infty \dfrac {n \sin n \pi} {\alpha^2 - n^2}
      | c = 
}}
{{eqn | r = \dfrac {\sin \alpha \pi} \alpha + 2 \sin \alpha \pi \sum_{n \mathop = 1}^\infty \dfrac {\alpha \cos n \pi } {\alpha^2 - n^2} - 0
      | c = $\sin n \pi = 0$
}}
{{eqn | n = 3
      | r = \dfrac {\sin \pi \alpha} \alpha + 2 \sin \pi \alpha \sum_{n \mathop \ge 1} \paren {-1}^n \dfrac \alpha {\alpha^2 - n^2} 
      | c = $\cos n \pi = \paren {-1}^n$
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \pi \cosec \pi \alpha
      | r = \dfrac 1 \alpha + 2 \sum_{n \mathop \ge 1} \paren {-1}^n \dfrac {\alpha} {\alpha^2 - n^2}
      | c = Mittag-Leffler Expansion for the Cosecant Function
}}
{{eqn | ll= \leadsto
      | l = \pi
      | r = \sin \pi \alpha \paren{\dfrac 1 \alpha + 2 \sum_{n \mathop \ge 1} \paren {-1}^n \dfrac {\alpha} {\alpha^2 - n^2} }
      | c = multiplying both sides by $\sin \pi \alpha$
}}
{{eqn | n = 4
      | r = \dfrac {\sin \pi \alpha} \alpha + 2 \sin \pi \alpha \sum_{n \mathop \ge 1} \paren {-1}^n \dfrac {\alpha} {\alpha^2 - n^2} 
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \pi
      | r = \sum_{n \mathop \in \Z} \dfrac {\map \sin {n + \alpha} \pi} {n + \alpha}
      | c = equating $(3)$ and $(4)$
}}
{{end-eqn}}
To establish this identity for all other values of $\theta$ on the interval $0 < \theta < 2 \pi$, we will demonstrate that the sum is a constant function.
We will do this by showing that the derivative of the function is zero everywhere which by Zero Derivative implies Constant Function will complete the proof.
We have:
{{begin-eqn}}
{{eqn | l = \map f \theta
      | r = \sum_{n \mathop \in \Z} \dfrac {\map \sin {n + \alpha} \theta} {n + \alpha}
      | c = 
}}
{{eqn | l = \map {f'} \theta
      | r = \sum_{n \mathop \in \Z} \map \cos {n + \alpha} \theta
      | c = Derivative of Sine Function/Corollary
}}
{{eqn | ll= \leadsto
      | l = \map {f'} \theta
      | r = \sum_{n \mathop \in \Z} \paren {\map \cos {\alpha \theta } \map \cos {n \theta} - \map \sin {\alpha \theta } \map \sin {n \theta} }
      | c = Cosine of Sum
}}
{{eqn | r = \map \cos {\alpha \theta} \sum_{n \mathop \in \Z} \map \cos {n \theta} - \map \sin {\alpha \theta} \sum_{n \mathop \in \Z} \map \sin {n \theta} 
      | c = Linear Combination of Indexed Summations
}}
{{end-eqn}}
From Cosine Function is Even and Sine Function is Odd, we have:
:$\map \cos {-n \theta} = \map \cos {n \theta}$
and:
:$\map \sin {-n \theta} = -\map \sin {n \theta}$
Therefore: 
{{begin-eqn}}
{{eqn | l = \map {f'} \theta
      | r = \map \cos {\alpha \theta} \paren {1 + 2 \sum_{n \mathop = 1}^\infty \map \cos {n \theta} } - \map \sin {\alpha \theta} \times 0
      | c = 
}}
{{eqn | r = \map \cos {\alpha \theta} \paren {1 + 2 \sum_{n \mathop = 1}^\infty \map \cos {n \theta} }
      | c = 
}}
{{eqn | r = \map \cos {\alpha \theta} \paren {1 + 2 \sum_{n \mathop = 1}^\infty \paren {\dfrac {e^{i n \theta} + e^{-i n \theta} } 2} }
      | c = Euler's Cosine Identity
}}
{{eqn | r = \map \cos {\alpha \theta} \paren {1 + \sum_{n \mathop = 1}^\infty e^{i n \theta} + \sum_{n \mathop = 1}^\infty e^{-i n \theta} }
      | c = Linear Combination of Indexed Summations
}}
{{eqn | r = \map \cos {\alpha \theta} \paren {\sum_{n \mathop = 0}^\infty e^{i n \theta} + \sum_{n \mathop = 0}^\infty e^{-i n \theta} - 1 }
      | c = Re-index the sum
}}
{{eqn | r = \map \cos {\alpha \theta} \paren {\dfrac 1 {1 - e^{i \theta} } + \dfrac 1 {1 - e^{-i \theta} } - 1}
      | c = Sum of Infinite Geometric Sequence
}}
{{eqn | r = \map \cos {\alpha \theta} \paren {\dfrac {\paren {1 - e^{-i \theta} } + \paren {1 - e^{i \theta} }  - \paren {1 - e^{-i \theta} } \paren {1 - e^{i \theta} } } {\paren {1 - e^{-i \theta} } \paren {1 - e^{i \theta} } } }
      | c = 
}}
{{eqn | r = \map \cos {\alpha \theta} \paren {\dfrac {\paren {1 - e^{-i \theta} } + \paren {1 - e^{i \theta} }  - \paren {1 - e^{i \theta} - e^{-i \theta} + 1} } {\paren {1 - e^{-i \theta} } \paren {1 - e^{i \theta} } } }
      | c = 
}}
{{eqn | r = 0
      | c = 
}}
{{end-eqn}}
{{qed}}