By defining several parameters, a half-discrete kernel function including its homogeneous and non-homogeneous forms is constructed. With the help of infinite series representation of tangent function and some techniques of analysis, a half-discrete Hilbert-type inequality with the constant factor expressed by cotangent function is established, and to prove that |\alpha {|}^{-\frac{\mathrm{1}}{q}}|\beta {|}^{-\frac{\mathrm{1}}{p}}\frac{\mathrm{\pi }}{\gamma }\left[\mathrm{\Phi }\left(\frac{\gamma \mathrm{\pi }}{{\lambda }_{\mathrm{1}}}\right)-\mathrm{\Phi }\left(\frac{\gamma \mathrm{\pi }}{{\lambda }_{\mathrm{2}}}\right)\right] is the optimal constant factor. In addition, by assigning the parameters different values, some special homogeneous and non-homogeneous Hilbert-type inequalities are established.