Verify for the Laplace transform of

$$F(t)=\{\begin{array}{ll}t& 0\le t<{t}_{0}\\ 2{t}_{0}-t& {t}_{0}\le t\le 2{t}_{0}\\ 0& t>2{t}_{0}\end{array}$$

$$\begin{array}{rl}\mathcal{L}\{F(t)\}& ={\int}_{0}^{{t}_{0}}t{e}^{-st}\text{}dt+{\int}_{{t}_{0}}^{2{t}_{0}}(2{t}_{0}-t){e}^{-st}\text{}dt\\ & ={\displaystyle \frac{1}{{s}^{2}}}{(1-{e}^{-s{t}_{0}})}^{2}\\ & ={\displaystyle \frac{4}{{s}^{2}}}{e}^{-s{t}_{0}}{\mathrm{sinh}}^{2}\left({\displaystyle \frac{1}{2}}s{t}_{0}\right)\end{array}$$

I was able to verify everything up to $\frac{1}{{s}^{2}}}{(1-{e}^{-s{t}_{0}})}^{2$, but I don't see how $\frac{1}{{s}^{2}}}{(1-{e}^{-s{t}_{0}})}^{2}={\displaystyle \frac{4}{{s}^{2}}}{e}^{-s{t}_{0}}{\mathrm{sinh}}^{2}\left({\displaystyle \frac{1}{2}}s{t}_{0}\right)$?