= Solution
For an open cover $\mathcal U=(U_i)$ and a sheaf $\mathcal F$, the <Čech cochain complex> is
$$
\check C^p(\mathcal U,\mathcal F)=\prod_{i_0<\cdots<i_p}\mathcal F(U_{i_0}\cap\cdots\cap U_{i_p}),
$$
with the alternating sum of restriction maps as differential. Its cohomology is the <Čech cohomology> $\check H^p(\mathcal U,\mathcal F)$.
Cover $\mathbb P_k^1$ by $U_0=D_+(x_0)$ and $U_1=D_+(x_1)$, and write $t=x_1/x_0$. This is an <acyclic cover theorem>[acyclic affine cover] for the <twisting sheaf on projective space> $\mathcal O(-1)$. In compatible trivializations, its degree-one Čech quotient is
$$
H^1(\mathbb P_k^1,\mathcal O(-1))
\cong
\frac{k[t,t^{-1}]}{k[t]+t^{-1}k[t^{-1}]}.
$$
The first summand contains every nonnegative power of $t$ and the second every negative power, so
$$
\boxed{H^1(\mathbb P_k^1,\mathcal O(-1))=0.}
$$
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