sh (x+t) - sh x = \frac{e^{x+t}-e^{-x-t}}{2} - \frac{e^x-e^{-x}}{2} = \frac{e^x \cdot e^t - e^{-x} \cdot e^{-t}-e^x+e^{-x}}{2} =
\frac{e^x \cdot (e^t-1) - e^{-x} \cdot (e^{-t}-1)}{2} = \\
= \frac{e^x \cdot (e^t-1) - e^{-x} (\frac{1}{e^t}-1)}{2} 
= \frac{e^x \cdot (e^t-1) - e^{-x} \cdot \frac{1-e^t}{e^t}}{2}
= \frac{\frac{e^t \cdot e^x \cdot (e^t-1) + e^{-x} \cdot (e^t-1)}{e^t}}{2} = \\
= \frac{(e^t-1)(e^{x+t}+e^{-x})}{2e^t}