Infinite Product of One Plus Reciprocals of Squares

 \[ \prod_{n = 1}^\infty \left( 1 + \frac{1}{n^2} \right) = \left( 1 + \frac{1}{1} \right) \times \left( 1 + \frac{1}{4} \right) \times \left( 1 + \frac{1}{9} \right) \times \cdots \] \[ = \frac{\sinh \pi}{\pi} = \frac{e^\pi - e^{-\pi}}{2\pi} \]

\[ \prod_{n = 1}^\infty \left( 1 + \frac{1}{n^2} \right) = \prod_{n = 1}^\infty \frac{n^2 + 1}{n^2} \] \[ = \prod_{n = 1}^\infty \frac{(n - i)(n + i)}{(n - 0)(n - 0)} \quad \text{(Difference of Two Squares)} \] \[ = \frac{\Gamma(1) \Gamma(1)}{\Gamma(1 + i) \Gamma(1 - i)} \quad \text{(Infinite Product Formula)} \]

\[ = \frac{\Gamma(1) \Gamma(1)}{i \Gamma(i) \Gamma(1 - i)} \quad \text{(Gamma Difference Equation)} \] \[ = \frac{0! \times 0! \sin(i\pi)}{i\pi} \quad \text{(Gamma Function Extends Factorial, Euler's Reflection Formula)} \] \[ = \frac{i \sinh{\pi}}{i\pi} \quad \text{(Hyperbolic Sine Definition)} \] \[ = \frac{\sinh{\pi}}{\pi} = \frac{e^\pi - e^{-\pi}}{2\pi} \]Από wikipedia: