= Solution
For a line bundle $\mathcal L$ on $X$, define the <homomorphism associated to a line bundle on an abelian variety>
$$
\phi_{\mathcal L}:X(k)\longrightarrow\operatorname{Pic}X,
\qquad
x\longmapsto T_x^*\mathcal L\otimes\mathcal L^\vee.
$$
The translation identity from part (i) gives
$$
\phi_{\mathcal L}(x+y)
\simeq\phi_{\mathcal L}(x)\otimes\phi_{\mathcal L}(y),
$$
so $\phi_{\mathcal L}$ is a homomorphism.
Suppose $\mathcal M\simeq T_y^*\mathcal L\otimes\mathcal L^\vee$ lies in its image. Then
$$
\phi_{\mathcal M}(x)
\simeq
T_{x+y}^*\mathcal L\otimes
(T_x^*\mathcal L)^\vee\otimes
(T_y^*\mathcal L)^\vee\otimes\mathcal L,
$$
which is trivial by the same translation identity. Hence $\phi_{\mathcal M}=0$; the image of every $\phi_{\mathcal L}$ lies in the <Identity component of the Picard group> $\operatorname{Pic}^0X$.
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