TITLE: Is $\sin(x)/ \tan(x) = \cos(x)$ at  $0$?
QUESTION [6 upvotes]: We know that $\frac{\sin(x)}{\tan(x)} = \cos(x)$. But at $x = 0$, the LHS becomes $0/0$. So is the function undefined at that point?

REPLY [3 votes]: The general convention for complex-valued functions of complex variable is: remove the removable discontinuities, e.g.
$$
\lim_{x\to 0} \frac{\sin x}{\tan x} = 1
$$
so take that to be the value of the function at 0.  (Except that it's conventional to use $z$ rather than $x$ here, and often one uses $x$ and $y$ for the real and imaginary parts of $z$, i.e. $z = x+iy$ and $x$ and $y$ are real.)
But could I add this: It seems to me that that that same convention frequently makes sense in trigonometry with no complex variables and no calculus to be seen anywhere.  And in trigonometry, it also usually makes sense to follow the same convention as in complex variables, that there is only one $\infty$, not a $+\infty$ and a $-\infty$.  That makes the tangent, cotangent, secant, and cosecant functions continuous everywhere.