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

Let $\alpha \in \R$ be a real number which is specifically not an integer.
For $0 \le \theta < 2 \pi$:
:$\ds \dfrac 1 \alpha + \sum_{n \mathop \ge 1} \dfrac {2 \alpha} {\alpha^2 - n^2} = \sum_{n \mathop \in \Z} \dfrac {\cos \paren {n + \alpha} \theta} {n + \alpha}$

Proof:
First we establish the following, as they will be needed later.
{{begin-eqn}}
{{eqn | o = 
      | r = \cos \paren {\alpha + n} \theta + \cos \paren {\alpha - n} \theta
      | c = 
}}
{{eqn | r = 2 \map \cos {\dfrac {\paren {\alpha + n} \theta + \paren {\alpha - n} \theta} 2} \map \cos {\dfrac {\paren {\alpha + n} \theta - \paren {\alpha - n} \theta} 2}
      | c = Cosine plus Cosine
}}
{{eqn | n = 1
      | r = 2 \cos \alpha \theta \cos n \theta
      | c = simplification
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | o = 
      | r = \cos \paren {\alpha + n} \theta - \cos \paren {\alpha - n} \theta
      | c = 
}}
{{eqn | r = -2 \map \sin {\dfrac {\paren {\alpha + n} \theta + \paren {\alpha - n} \theta} 2} \map \sin {\dfrac {\paren {\alpha + n} \theta - \paren {\alpha - n} \theta} 2}
      | c = Cosine minus Cosine
}}
{{eqn | n = 2
      | r = -2 \sin \alpha \theta \sin n \theta
      | c = simplification
}}
{{end-eqn}}
We have:
{{begin-eqn}}
{{eqn | o = 
      | r = \sum_{n \mathop = -\infty}^\infty \dfrac {\cos \paren {n + \alpha} \theta} {n + \alpha}
      | c = 
}}
{{eqn | r = \dfrac {\cos \paren {0 + \alpha} \theta} {0 + \alpha} + \sum_{n \mathop = 1}^\infty \dfrac {\cos \paren {n + \alpha} \theta} {n + \alpha} + \sum_{n \mathop = -\infty}^{-1} \dfrac {\cos \paren {n + \alpha} \theta} {n + \alpha}
      | c = 
}}
{{eqn | r = \dfrac {\cos \alpha \theta} \alpha + \sum_{n \mathop = 1}^\infty \dfrac {\cos \paren {n + \alpha} \theta} {n + \alpha} + \sum_{n \mathop = 1}^\infty \dfrac {\cos \paren {-n + \alpha} \theta} {-n + \alpha}
      | c = 
}}
{{eqn | r = \dfrac {\cos \alpha \theta} \alpha + \sum_{n \mathop = 1}^\infty \paren {\dfrac {\cos \paren {\alpha + n} \theta} {\alpha + n} + \dfrac {\cos \paren {\alpha - n} \theta} {\alpha - n} }
      | c = 
}}
{{eqn | r = \dfrac {\cos \alpha \theta} \alpha + \sum_{n \mathop = 1}^\infty \paren {\dfrac {\paren {\alpha - n} \paren {\cos \paren {\alpha + n} \theta} + \paren {\alpha + n} \paren {\cos \paren {\alpha - n} \theta} }{\alpha^2 - n^2} }
      | c = 
}}
{{eqn | r = \dfrac {\cos \alpha \theta} \alpha + \sum_{n \mathop = 1}^\infty \paren {\dfrac {\alpha \cos \paren {\alpha + n} \theta - n \cos \paren {\alpha + n} \theta + \alpha \cos \paren {\alpha - n} \theta + n \cos \paren {\alpha - n} \theta} {\alpha^2 - n^2} }
      | c = 
}}
{{eqn | r = \dfrac {\cos \alpha \theta} \alpha + \sum_{n \mathop = 1}^\infty \paren {\dfrac {\alpha \paren {\cos \paren {\alpha + n} \theta + \cos \paren {\alpha - n} \theta} - n \paren {\cos \paren {\alpha + n} \theta - \cos \paren {\alpha - n} \theta} } {\alpha^2 - n^2} }
      | c = 
}}
{{eqn | r = \dfrac {\cos \alpha \theta} \alpha + \sum_{n \mathop = 1}^\infty \paren {\dfrac {2 \alpha \cos \alpha \theta \cos n \theta + 2 n \sin \alpha \theta \sin n \theta} {\alpha^2 - n^2} }
      | c = from $(1)$ and $(2)$
}}
{{eqn | r = \dfrac {\cos \alpha \theta} \alpha + 2 \alpha \cos \alpha \theta \sum_{n \mathop = 1}^\infty \paren {\dfrac {\cos n \theta} {\alpha^2 - n^2} } + 2 \sin \alpha \theta \sum_{n \mathop = 1}^\infty \paren {\dfrac {n \sin n \theta} {\alpha^2 - n^2} }
      | c = Linear Combination of Indexed Summations
}}
{{end-eqn}}
Setting $\theta = 0$:
{{begin-eqn}}
{{eqn | l = \sum_{n \mathop = -\infty}^\infty \dfrac {\map \cos {\paren {n + \alpha} 0} } {n + \alpha}
      | r = \dfrac {\cos \alpha 0} \alpha + 2 \alpha \cos \alpha 0 \sum_{n \mathop = 1}^\infty \paren {\dfrac {\cos n 0 } {\alpha^2 - n^2} } + 2 \sin \alpha 0 \sum_{n \mathop = 1}^\infty \paren {\dfrac {n \sin n 0} {\alpha^2 - n^2} }
      | c = 
}}
{{eqn | r = \dfrac 1 \alpha + \sum_{n \mathop = 1}^\infty \dfrac {2 \alpha} {\alpha^2 - n^2} + 0
      | c = Sine of Zero is Zero and Cosine of Zero is One
}}
{{eqn | r = \dfrac 1 \alpha + \sum_{n \mathop = 1}^\infty \dfrac {2 \alpha} {\alpha^2 - n^2}
      | c = 
}}
{{end-eqn}}
To establish this identity for all other values of $\theta$ on the interval $0 \le \theta < 2\pi$, 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 \cos {n + \alpha} \theta} {n + \alpha}
      | c = 
}}
{{eqn | l = \map {f'} \theta
      | r = \sum_{n \mathop \in \Z} -\map \sin {n + \alpha} \theta
      | c = Derivative of Cosine Function/Corollary
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \map {f'} \theta
      | r = -\sum_{n \mathop \in \Z} \paren {\map \sin {\alpha \theta} \map \cos {n \theta} + \map \cos {\alpha \theta } \map \sin {n \theta} }
      | c = Sine of Sum
}}
{{eqn | r = -\map \sin {\alpha \theta } \sum_{n \mathop \in \Z} \map \cos {n \theta} - \map \cos {\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 \sin {\alpha \theta} \paren {1 + 2 \sum_{n \mathop = 1}^\infty \map \cos {n \theta} } - \map \cos {\alpha \theta} \times 0
      | c = 
}}
{{eqn | r = -\map \sin {\alpha \theta} \paren {1 + 2 \sum_{n \mathop = 1}^\infty \map \cos {n \theta} }
      | c = 
}}
{{eqn | r = -\map \sin {\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 \sin {\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 \sin {\alpha \theta} \paren {\sum_{n \mathop = 0}^\infty e^{i n \theta} + \sum_{n \mathop = 0}^\infty e^{-i n \theta} - 1}
      | c = re-indexing the sum
}}
{{eqn | r = -\map \sin {\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 \sin {\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 \sin {\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}}