Picard group and cohomology
Solution 1:
Suppose for simplicity that $X$ is integral. Consider the exact sequence of groups $$ 1\to O_X^* \to K_X^* \to K_X^*/O_X^* \to 1$$ where $K_X$ is the constant sheaf of rational functions on $X$. Taking the cohomology will give $$ K(X)^* \to H^0(X, K_X^*/O_X^*) \to H^1(X, O_X^*) \to H^1(X, K_X^*).$$ Now the last term vanishes because $K_X^*$ is a flasque sheaf, and the cokernel of the left arrow is by definition the group of Cartier divisors on $X$ up to linear equivalence. As $X$ is integral, this cokernel is known to be isomorphic to $\mathrm{Pic}(X)$.