  Based on $10.64~\mathrm{fb}^{-1}$ of $e^+e^-$ collision data taken at
center-of-mass energies between 4.237 and 4.699 GeV with the BESIII detector,
we study the leptonic $D^+_s$ decays using the $e^+e^-\to D^{*+}_{s}
D^{*-}_{s}$ process. The branching fractions of
$D_s^+\to\ell^+\nu_{\ell}\,(\ell=\mu,\tau)$ are measured to be
$\mathcal{B}(D_s^+\to\mu^+\nu_\mu)=(0.547\pm0.026_{\rm stat}\pm0.016_{\rm
syst})\%$ and $\mathcal{B}(D_s^+\to\tau^+\nu_\tau)=(5.60\pm0.16_{\rm
stat}\pm0.20_{\rm syst})\%$, respectively. The product of the decay constant
and Cabibbo-Kobayashi-Maskawa matrix element $|V_{cs}|$ is determined to be
$f_{D_s^+}|V_{cs}|=(246.5\pm5.9_{\rm stat}\pm3.6_{\rm syst}\pm0.5_{\rm
input})_{\mu\nu}~\mathrm{MeV}$ and $f_{D_s^+}|V_{cs}|=(252.7\pm3.6_{\rm
stat}\pm4.5_{\rm syst}\pm0.6_{\rm input}))_{\tau \nu}~\mathrm{MeV}$,
respectively. Taking the value of $|V_{cs}|$ from a global fit in the Standard
Model, we obtain ${f_{D^+_s}}=(252.8\pm6.0_{\rm stat}\pm3.7_{\rm
syst}\pm0.6_{\rm input})_{\mu\nu}$ MeV and ${f_{D^+_s}}=(259.2\pm3.6_{\rm
stat}\pm4.5_{\rm syst}\pm0.6_{\rm input})_{\tau \nu}$ MeV, respectively.
Conversely, taking the value for $f_{D_s^+}$ from the latest lattice quantum
chromodynamics calculation, we obtain $|V_{cs}| =(0.986\pm0.023_{\rm
stat}\pm0.014_{\rm syst}\pm0.003_{\rm input})_{\mu\nu}$ and $|V_{cs}| =
(1.011\pm0.014_{\rm stat}\pm0.018_{\rm syst}\pm0.003_{\rm input})_{\tau \nu}$,
respectively.
