= Solution
Cover the <projective line> by $U_0=\operatorname{Spec}k[t]$ and $U_1=\operatorname{Spec}k[t^{-1}]$. On their overlap the coordinate is invertible. Choose frames $e_0,e_1$ of the <twisting sheaf on projective space> $\mathcal O(n)$ with $e_1=t^ne_0$. The <vanishing of quasi-coherent cohomology on an affine scheme> applies to both charts and their intersection, so the <Čech cochain complex>
$$
0\longrightarrow k[t]\oplus k[t^{-1}]\xrightarrow{\delta}k[t,t^{-1}]\longrightarrow0,\qquad\delta(a,b)=t^nb-a,
$$
computes the <sheaf cohomology>. Its kernel identifies with $k[t]\cap t^nk[t^{-1}]$, and its cokernel is
$$
\frac{k[t,t^{-1}]}{k[t]+t^nk[t^{-1}]}.
$$
For $n\geq0$, the intersection has basis $1,t,\ldots,t^n$; for $n<0$ it is zero. In the quotient, powers $t^j$ with $j\geq0$ or $j\leq n$ disappear. The surviving basis for $n\leq-2$ is $t^{n+1},\ldots,t^{-1}$; for $n\geq-1$ none survives. Hence
$$
\boxed{h^0(\mathcal O(n))=\max(n+1,0),\qquad h^1(\mathcal O(n))=\max(-n-1,0),\qquad H^i(\mathcal O(n))=0\ (i\geq2).}
$$
This is the <Čech cohomology of twists on the projective line>.
By the preceding question, $\omega=\Omega^1_{\mathbb P^1/k}=\mathcal O(-2)$. In the $dt$ frame its degree-one <Čech cohomology> is one-dimensional, represented by $t^{-1}dt$. Define
$$
\operatorname{tr}:H^1(\omega)\longrightarrow k,\qquad [h(t)dt]\longmapsto[t^{-1}]h(t).
$$
This is well defined: differentials regular on $U_0$ have nonnegative powers, and those regular on $U_1$ have powers at most $-2$ when expressed using $dt$. Evaluation and the <cup product> give natural pairings
$$
H^i(E)\times H^{1-i}(E^\vee\otimes\omega)\longrightarrow H^1(\omega)\xrightarrow{\operatorname{tr}}k,\qquad i=0,1.
$$
For $E=\mathcal O(n)$ with $n\geq0$, the basis element $t^a$ in $H^0(E)$ pairs with $t^{-a-1}dt$ in $H^1(E^\vee\otimes\omega)$ and with no other element of this dual Laurent basis. The pairing matrix is a permutation matrix, hence invertible. For $n\leq-2$ the same computation exchanges degrees zero and one. For $n=-1$ both sides vanish. Thus the pairings are perfect for every <line bundle> on $\mathbb P^1$.
Here we used the classification of <line bundles> on the <projective line>: they are $\mathcal O(a)$ for integers $a$. One direct justification is that every <line bundle> has a nonzero rational section and hence is associated with a <divisor on an algebraic curve>. The finite closed point defined by an irreducible polynomial $p(t)$ is linearly equivalent to $(\deg p)\infty$, using the <principal divisor> of $p(t)$. Thus every divisor is equivalent to its degree times $\infty$, proving the classification over arbitrary $k$.
To extend the pairing to every <locally free coherent sheaf> $E$, induct on rank, without assuming the splitting theorem of the next question. A one-dimensional subspace of the generic fibre, saturated in $E$, gives an exact sequence
$$
0\longrightarrow L\longrightarrow E\longrightarrow Q\longrightarrow0
$$
with $L$ a <line bundle> and $Q$ a lower-rank <vector bundle>. This <saturated line subbundle on a smooth curve> exists because its stalk and its torsion-free quotient are free over the <discrete valuation rings> of the curve. The dual sequence, tensored with $\omega$, is exact as well, since the original sequence splits locally.
The two <long exact sequences in sheaf cohomology>, after dualizing one, are compatible with the evaluation pairings. In <Čech cohomology>, this compatibility follows by lifting a section locally: the differences of its lifts represent its connecting class, and contracting those differences with a dual cocycle gives the transpose connecting map, with the usual sign. Thus the connecting maps are adjoint; changing one vertical map's sign gives a commuting diagram. Its rows are
$$
\begin{aligned}
0&\to H^0(L)\to H^0(E)\to H^0(Q)\to H^1(L)\to H^1(E)\to H^1(Q)\to0,\\
0&\to H^1(L^\vee\omega)^*\to H^1(E^\vee\omega)^*\to H^1(Q^\vee\omega)^*\to H^0(L^\vee\omega)^*\to H^0(E^\vee\omega)^*\to H^0(Q^\vee\omega)^*\to0.
\end{aligned}
$$
The maps for $L$ and $Q$ are isomorphisms by the line-bundle calculation and induction. Exactness then makes the maps for $E$ isomorphisms, by the <Five lemma> (or by comparing the relevant kernels and cokernels). All groups are finite-dimensional because the curve is projective and the sheaves coherent. Therefore <residue duality on the projective line> proves
$$
\boxed{H^i(\mathbb P^1,E)\cong H^{1-i}(\mathbb P^1,E^\vee(-2))^*,\qquad i=0,1.}
$$
These isomorphisms are natural in $E$. Higher <sheaf cohomology> vanishes by the same two-chart complex.
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