Why maple computes $\prod_{k=3}^{\infty} \cos \left(\frac{\pi}{k}\right)$ as $0$? I have computed the infinite multiplication using Maple, $\Pi_{k=3}^{\infty} ( \cos (\frac{\pi}{k} ))$, as follows, but it resulted in 0! I wonder why this happened, although maple was using exact arithmetic.
P := Product(cos(Pi/k), k = 3 .. infinity)
value(P)

Note that if I use floating-point instead, it gives me the right answer, 0.1149420449.
evalf(P)

By the way, I did not expect such a situation! Exact arithmetic should be exact! I am multiplying non-zero numbers to each other, starting from $1/2$ to $1$ as $n \rightarrow \infty$. So it should not be zero! Why such a thing happened?
Note: Maple is a proprietary Computer Algebraic System that I have access to through my university's lab.
 A: Convergence issues are quite commonly bad handled by CASs. 
I remember older versions of Mathematica giving
$$\sum_{n=0}^{+\infty}\cos(n\pi) = \frac{1}{2},$$
for istance. Anyway, from the Weierstrass product for the cosine function, namely:
$$\cos(z) = \prod_{m=0}^{+\infty}\left(1-\frac{4z^2}{\pi^2(2m+1)^2}\right)\tag{1}$$
it follows that:
$$\log\cos\frac{\pi}{k}=\sum_{m=0}^{+\infty}\log\left(1-\frac{4}{k^2(2m+1)^2}\right)=-\sum_{m=0}^{+\infty}\sum_{n=1}^{+\infty}\frac{1}{n}\cdot\frac{4^n}{k^{2n}(2m+1)^{2n}},\tag{2}$$
and summing over $m$ we get:
$$\log\cos\frac{\pi}{k}=-\sum_{n=1}^{+\infty}\frac{(4^n-1)\cdot\zeta(2n)}{n\cdot k^{2n}}\tag{3}.$$
Now summing both sides of $(3)$ with $k$ that goes from $3$ to $+\infty$ gives:
$$\prod_{k=3}^{+\infty}\cos\frac{\pi}{k}=\exp\left(-\sum_{n=1}^{+\infty}\frac{1}{n}(4^n-1)\cdot\zeta(2n)\cdot\left(\zeta(2n)-1-4^{-n}\right)\right)\tag{4}$$
where the terms of the sum in the RHS behaves like $(4/9)^n$ ensuring pretty fast convergence. Is is also quite interesting to notice that, by the Euler reflection formula for the Gamma function,
$$\sum_{n=1}^{+\infty}\frac{4^n}{d^{2n}}\cdot\frac{\zeta(2n)}{n}=-\log\left(\frac{d}{2\pi}\sin\frac{2\pi}{d}\right).\tag{5}$$
A: OK! I have asked this question on several other forums and the following is the summary of the responses and shall be assumed as answer.
The reason that this happens in Maple is a solving the equation $\cos(\Pi/k)=0$ and finding all of its integer roots in $\left[ 3 , \infty \right)$ to see if it is zero or not ($product$ is some sort of a smart function in maple, rather than trying to multiply the numbers in order, it first checks for short circuits). The solution to this equation would be of the general form $k=\frac{2}{1+2*i}$ for integer $i$. But, Maple has a $3$-valued logic, so the value of a logical expression could be either true, false, and Undefined! The developer of $Product$ unfortunately seems to forget this, Maple is closed-source and no access to the source code, and ruined everything!
P.S.: This answer is a summary of the same question asked in www.mapleprimes.com by me.
