Infinite Products -- Tangent function?

Question

I've been looking around and I see no formulas given in any of the sources I've been able to find for the infinite product representing $\tan\left(x\right)$. Is it simply the ratio of the infinite products for $\sin\left(x\right)$ and $\cos\left(x\right)$? In which case, it should look like $$ \frac{x\cdot\prod\limits_{n=1}^{\infty}\left(1-\frac{x^2}{n^2\pi^2}\right)}{\prod\limits_{n=1}^{\infty}\left(1-\frac{4x^2}{\pi^2\left(2n-1\right)^2}\right)}=\frac{\sin\left(x\right)}{\cos\left(x\right)}=\tan\left(x\right)? $$ Forgive me if there is an obvious solution to this problem that I haven't seen as of yet. I've taken up to differential equations and multivariable calc so far in my major.

Brandon

Answer 1

• The upper product should start at $n=1$, otherwise you have division by $0$.
• The x before the infinite product in the numerator should be changed to $\pi x$.
• The arguments of the sine, cosine, and tangent functions should be $\pi x$ rather than x. Unless you want to imply that $~\sin(0)=\sin(\pm1)=\sin(\pm2)=\sin(\pm3)=\ldots=0$, as opposed to $\sin(0)=\sin(\pm\pi)=\sin(\pm2\pi)=\sin(\pm3\pi)=\ldots=0$.