TITLE: is sin(x) / sin(x) = 1?
QUESTION [7 upvotes]: I tested this in python using:
import numpy as np
import matplotlib.pyplot as plt

x = np.linspace(0, 10*2*np.pi, 10000)
y = np.sin(x)
plt.plot(y/y)
plt.plot(y)

Which produces:

The blue line representing sin(x)/sin(x) appears to be y=1
However, I don't know if the values at the point where sin(x) crosses the x-axis really equals 1, 0, infinity or just undefined.

REPLY [3 votes]: There are few misconceptions regarding the rational functions. For example, if
$$
f(x) = \frac{x^2 -1}{x-1} 
$$
Then, we usually find that people write it out as
$$
f(x) = x+1 
$$
But they are not equivalent, they differ from each other at $x=1$ where the former is undefined while the latter have the value as $1$. Let's start with $f(x) = x+1$ and try to get $\frac{x^2 -1}{x-1}$.
$$
f(x) = x+1 \\
f(x) = x+1 \times \frac{x-1}{x-1} \\
f(x)= \frac{x^2-1}{x-1}
$$
We can justify the second step by saying "well, $\frac{x-1}{x-1}$ is basically 1, we got a division by itself" but we forget two things, first $x-1$ is not a constant like real numbers it's a changing quantity, second the at $x=1$ we will get $x-1$ as 0. Now, will the argument "we got a division by itself" work? No, because 0 is something which doesn't have a property "itself". You can read division by zero here
So, in the case $\frac{\sin x}{\sin x}$ we get $\sin x$ as 0 for $x= 2\pi n $, $n= 0, 1 \cdots $. Therefore, functions
$$
f(x) = \frac{\sin x}{\sin x} \\
g(x) =1\\
$$
are equivalent, except at points $x=0, 2\pi, 4\pi, 6\pi \cdots$.
Hope it helps!