Why is y=(sin(2x))/(2sinx) only undefined at the y-intercept, and not for all pi values after?

sin2x can be written as 2sinx*cosx, so sin2x/(2sinx) → cosx 

(this is, in fact, defined at x = 0)

.

It isn't defined at any multiple of  pi
 

sin(2x) / [2sin(x)], for x =  0   →   undefined

sin(2x) / [2sin(x)], for x =  pi   →   undefined      .....etc....

You can check this here, by evaluating x for any pi multiple : 

https://www.wolframalpha.com/input/?i=sin(2x)+%2F+%5B2sin(x)%5D,+for+x%C2%A0%3D+0