This question comes from Silverman's AEC, exercise 2.11, self-study. The goal is to prove Weil reciprocity: if $C$ is a smooth curve over $K$ and $f,g$ are nonzero functions in the function field with disjoint supports, then
$$f(\operatorname{div} g) = g(\operatorname{div} f)$$
where for a divisor $D = \sum_P n_P P$,
$$f(D):= \prod_P f(P)^{n_P}$$
Now, we are told to first prove this for $\mathbb P^1$, which is easy to do. We are then asked to prove it for general $C$ by considering $f,g$ as maps $f, g: C \to \mathbb P^1$ to reduce to the above case. Here is all I have been able to do thus far, and it seems to lead to a mess of calculations quickly instead of reducing the problem to a simpler case. Let $\cdot^*$ indicate pullback of function fields, and let $\cdot_*$ denote pushforward, which is the inverse of pullback composed with the norm map. Let $i : \mathbb P^1 \to \mathbb P^1$ the identity map. Then, using established properties of these operations:
$$f(\operatorname{div}g) = f(g^* \operatorname{div} i) = (g_* f)\operatorname{div} i = (g_* f)(0 - \infty) = (g_*f)(0)\cdot(g_*f)(\infty)^{-1} = (g^{*-1} \circ N(f))(0) \cdot (g^{*-1} \circ N(f))(\infty)^{-1} $$
From here, if my extension of function fields is Galois, I can replace $N(f)$ by $\prod_\sigma \sigma(f)$. From there, I am stuck. This doesn't seem to be reducing the problem down.
Here are two sources I looked at:
https://math.mit.edu/classes/18.782/ProblemSet9.pdf
which seems to indicate the result should follow simply from known results about pushforward, pullback and divisors, and
http://www.math.uwaterloo.ca/~djao/co690.2007/weil.pdf
whose Theorem 2.1 seems false and unfamiliar (I think it should say that a divisor on an elliptic curve is principal iff its degree is $0$, although one direction of that is not clear to me, and whose approach seems more general than intended.