= Solution
For $U_{i_0\dots i_q}=U_{i_0}\cap\cdots\cap U_{i_q}$, the <Čech cochain complex> has groups
$$
C^q(\mathcal U,\mathcal F)=\prod_{i_0<\cdots<i_q}\mathcal F(U_{i_0\dots i_q}),\qquad
(\delta c)_{i_0\dots i_{q+1}}=\sum_{j=0}^{q+1}(-1)^j c_{i_0\dots\widehat{i_j}\dots i_{q+1}}|_{U_{i_0\dots i_{q+1}}}.
$$
Terms in $\delta^2$ cancel in pairs, and <Čech cohomology> is $\ker\delta/\operatorname{im}\delta$. A degree-zero <cocycle> is exactly a family of compatible local sections. The <sheaf gluing axiom> gives their unique <global section>, proving $\check H^0(\mathcal U,\mathcal F)\cong\mathcal F(X)$. If the cover has affine finite intersections, a <quasi-coherent sheaf> has no higher <cohomology> on those intersections by <vanishing of quasi-coherent cohomology on an affine scheme>. The <acyclic cover theorem> then identifies all Čech groups with <sheaf cohomology>. In particular, a finite affine <open cover> of a <separated variety> has this property.
For the <sheaf of units of the structure sheaf>, a multiplicative degree-one <cocycle> is a family $g_{ij}\in\mathcal O_X^*(U_i\cap U_j)$ satisfying $g_{ij}g_{jk}=g_{ik}$, with $g_{ji}=g_{ij}^{-1}$. It glues trivial rank-one <free modules> into an <invertible sheaf>. Changing the local frames multiplies $g_{ij}$ by a <coboundary>, and two sets of transition data give isomorphic <line bundles> precisely when their <cocycles> differ this way. Tensoring <line bundles> multiplies their <cocycles>. Thus <line bundles trivialized by an open cover> give the group <isomorphism>
$$
\operatorname{Pic}(X)_{\mathcal U}\cong\check H^1(\mathcal U,\mathcal O_X^*).
$$
For the remaining arguments, work over the algebraically closed ground field. On the <irreducible variety> $V$, put $\mathcal Q=\mathcal K^*/\mathcal O_V^*$, a quotient of <sheaves of abelian groups>. A <global section> of $\mathcal Q$ is locally represented by <rational functions> $f_i$ whose ratios are regular units. The corresponding unit <cocycle> $f_j/f_i$ defines an <invertible sheaf>. A single global <rational function> has trivial <cocycle>. Conversely, every <line bundle> has a nonzero rational section: choose a nonzero vector in its one-dimensional fibre at the <generic point> and express it in local frames. This supplies such local $f_i$. If the associated <line bundle> is trivial, changing frames makes all $f_i$ restrictions of one <rational function>. Therefore
$$
k(V)^*\longrightarrow\Gamma(V,\mathcal K^*/\mathcal O_V^*)\longrightarrow\operatorname{Pic}(V)\longrightarrow0
$$
is exact. This is the <Cartier-divisor description of the Picard group>. The <sheaf of nonzero rational functions on an irreducible variety> is <flasque>, since all restrictions between nonempty <open sets> are the identity on $k(V)^*$. Apply the <long exact sequence in sheaf cohomology> to $1\to\mathcal O_V^*\to\mathcal K^*\to\mathcal Q\to1$. Since $H^1(V,\mathcal K^*)=0$, its connecting map has exactly the <cokernel> just computed, proving
$$
\boxed{\operatorname{Pic}(V)\cong H^1(V,\mathcal O_V^*).}
$$
Finally the <Segre description of a smooth quadric surface> identifies $V$ with $\mathbb P^1\times\mathbb P^1$. The two <rulings of a smooth quadric surface> have classes $F_1,F_2$ generating $\operatorname{Pic}(V)\cong\mathbb Z^2$. For completeness, remove one line in each ruling: the remaining chart is the <affine plane>, with factorial <coordinate ring> $k[s,t]$ and trivial <divisor class group>. The <localization sequence for the divisor class group> makes $F_1,F_2$ generators, and their degrees on the two ruling lines prove independence. The hyperplane class, and hence the conic $C$, has bidegree $(1,1)$. By <Picard-group localization on a smooth variety>, the <Picard group of a smooth affine quadric surface> is
$$
\boxed{\operatorname{Pic}(U)\cong\mathbb Z^2/\mathbb Z(1,1)\cong\mathbb Z.}
$$
Explicitly, the restriction of $\mathcal O_V(1,0)$ is nontrivial: if it were trivial on $U$, its rational trivialization would have divisor supported on $C$, forcing $(1,0)$ to be an integer multiple of $(1,1)$, which is impossible.
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