prove-that-cos-piz-n-1-1-z-2-1-2-n-2-

Question Number 67531 by mathmax by abdo last updated on 28/Aug/19

p r o v e t h a t c o s (π z) = \prod_{n = 1}^{\infty} (1 - \frac{z^{2}}{{(\frac{1}{2} + n)}^{2}})

Commented by ~ À ® @ 237 ~ last updated on 29/Aug/19

K n o w i n g c o s x = \frac{s i n 2 x}{2 s i n x}

c o s (π z) = \frac{s i n (2 π z)}{2 s i n (π z)}

N o w u s i n g s i n (π z) = π z \underset{n = 1}{\prod^{\infty}} (1 - \frac{z^{2}}{n^{2}})

c o s (π z) = \frac{(2 π z) \underset{n = 1}{\prod^{\infty}} (1 - \frac{{(2 z)}^{2}}{n^{2}})}{2 (π z) \underset{n = 1}{\prod^{\infty}} (1 - \frac{z^{2}}{n^{2}})} = \frac{\underset{n = 1}{\prod^{\infty}} (1 - \frac{4 z^{2}}{{(2 n)}^{2}}) \underset{n = 1}{\prod^{\infty}} (1 - \frac{4 z^{2}}{{(2 n + 1)}^{2}})}{\underset{n = 1}{\prod^{\infty}} (1 - \frac{z^{2}}{n^{2}})} = \underset{n = 1}{\prod^{\infty}} (1 - \frac{z^{2}}{{(\frac{2 n + 1}{2})}^{2}})