= Solution
A <holomorphic line bundle> $L\to X$ is a rank-one <holomorphic vector bundle>: it has local trivializations $L|_{U_i}\cong U_i\times\mathbb C$ whose fiberwise linear <transition functions of a vector bundle> are nowhere-zero <holomorphic functions>. Equivalently, choose local holomorphic frames $e_i$ and write $e_j=g_{ij}e_i$ on overlaps, with $g_{ij}\in\mathcal O^*(U_i\cap U_j)$ and the cocycle relation on triple overlaps. A <holomorphic section> over $U$ has coefficients $s_i$ holomorphic on $U\cap U_i$ in $s=s_i e_i$; consistency means $s_i=g_{ij}s_j$.
The <dual bundle> has fiber $L_x^*=\operatorname{Hom}_{\mathbb C}(L_x,\mathbb C)$. Its dual frames $e_i^*$ are characterized by $e_i^*(e_i)=1$, and the frame relation becomes $e_j^*=g_{ij}^{-1}e_i^*$. Reciprocals of nowhere-zero <holomorphic functions> are holomorphic, and their cocycle relations persist. This defines the <holomorphic dual vector bundle> structure on $L^*$, with holomorphic evaluation pairing $L^*\otimes L\to\mathcal O_X$.
Assume now that $X$ is compact and connected, and let $s$ and $t$ be nonzero global <holomorphic sections> of $L$ and $L^*$, respectively. The function $t(s)$ is a global <holomorphic function>. The <maximum modulus principle>, applied in local complex coordinate lines near a maximum and then propagated on the connected manifold, makes it constant. Moreover it is not identically zero: the zero locus of a nonzero <holomorphic section> has empty interior by the <identity theorem>, so the two nonvanishing loci are dense open sets and intersect. Thus $t(s)=c\ne0$ everywhere.
In particular, $s$ never vanishes. The map $(x,a)\mapsto a s(x)$ is a holomorphic bundle isomorphism $X\times\mathbb C\to L$; in a local frame its inverse divides by the nowhere-zero holomorphic coefficient of $s$. Conversely, a holomorphic trivialization supplies the constant unit sections of both $L$ and $L^*$. We have proved the <two-section criterion for holomorphic triviality>:
$$
\boxed{L\text{ is holomorphically trivial}\quad\Longleftrightarrow\quad
H^0(X,L)\ne0\text{ and }H^0(X,L^*)\ne0.}
$$
Write points of <Complex projective space> as lines $[Z]\subset\mathbb C^{n+1}$. The <complex tautological line bundle> is
$$
\mathcal O(-1)=\{([Z],v):v\in\mathbb C Z\}\subset\mathbb P^n\times\mathbb C^{n+1},
$$
with fiber that very line. Its dual is the <hyperplane line bundle> $\mathcal O(1)$. On the affine chart $U_i=\{Z_i\ne0\}$, the vector $e_i([Z])=Z/Z_i$ is a nowhere-zero holomorphic frame of $\mathcal O(-1)$. On $U_i\cap U_j$,
$$
e_j=\frac{Z_i}{Z_j}e_i,\qquad
e_j^*=\frac{Z_j}{Z_i}e_i^*.
$$
Both ratios are nowhere-zero <holomorphic functions> of the affine coordinates. They exhibit the holomorphic <transition functions of a vector bundle> for the two bundles explicitly.
Choose a nonzero linear functional $\ell:\mathbb C^{n+1}\to\mathbb C$ whose zero hyperplane is $H$. At $[Z]$, its restriction to $\mathbb C Z$ is an element of that line's dual and therefore defines a <holomorphic section> $s_\ell$ of $\mathcal O(1)$. In the dual frame above its coefficient is
$$
s_\ell=\frac{\ell(Z)}{Z_i}e_i^*\quad\text{on }U_i.
$$
This expression is holomorphic even along $H$, and its zero locus is exactly $H$. Consequently \b[$s_\ell|_{\mathbb P^n\setminus H}$ is the requested nowhere-zero section; its global holomorphic extension vanishes precisely on $H$]. The extension need not, and does not, stay nowhere zero on the hyperplane.
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