the interchange of zeros and poles under 1/f
Zeros and poles are mirror images of each other, and the mirror is the operation of taking a reciprocal. Where a function f touches zero, its reciprocal 1/f shoots to infinity; where f shoots to infinity, 1/f touches zero. Forming 1/f swaps the two kinds of special point, and — this is the precise and useful part — it swaps them keeping the SAME order.
Here is the exact rule. If f has a zero of order m at z_0 (meaning f(z) = (z - z_0)^m times a holomorphic function that is nonzero at z_0), then 1/f has a pole of order m at z_0. Conversely, if f has a pole of order m at z_0, then 1/f has a zero of order m there. The orders match because near the point f looks like c(z - z_0)^m (with c not zero) for a zero, so 1/f looks like (1/c)(z - z_0)^(-m) — a pole of exactly order m — and the same algebra runs in reverse. A simple zero becomes a simple pole; a double zero becomes a double pole.
This interchange is the engine behind how meromorphic functions are built and analyzed. It explains why the poles of a rational function g/h sit exactly at the zeros of h (those not cancelled by zeros of g), why tan z = sin z / cos z has its poles where cos z vanishes, and why the argument principle can count zeros and poles together with one integral: under the logarithmic derivative, a zero of order m and a pole of order m contribute opposite signs, +m and -m. Once you see zeros and poles as two faces of one coin, much of meromorphic theory becomes symmetric.
f(z) = sin z has a simple zero at 0 (sin z = z - z^3/6 + ...). Then 1/sin z = 1/(z(1 - z^2/6 + ...)) = (1/z)(1 + z^2/6 + ...) has a simple pole at 0, residue 1. The order-1 zero of sin became the order-1 pole of 1/sin.
A simple zero of sin z at 0 turns into a simple pole of 1/sin z with residue 1.
The rule needs the point to be an ISOLATED zero (or pole) of finite order; if f is identically zero on a neighbourhood, 1/f is not even defined, and an essential singularity of f does not turn into a pole of 1/f.