Consider the quantity $\sinh (\phi_1)\cosh (\phi_2) + \cosh (\phi_1)\sinh (\phi_2)$. By expressing each term as an exponential ($\sinh (\phi)=(e^{\phi} – e^{-\phi})/2$, $\cosh (\phi)=(e^{\phi} + e^{-\phi})/2$), simplify the quantity to one of the following four results.

Consider the quantity $\cosh (\phi_1)\cosh (\phi_2) + \sinh (\phi_1)\sinh (\phi_2)$. By expressing each term as an exponential, simplify the quantity to one of the following four results.

Consider $\tanh (\phi_1 + \phi_2)={\sinh (\phi_1 + \phi_2) \over \cosh (\phi_1 + \phi_2)}$. Using the results of the previous parts of this problem, express this in terms of hyperbolic trigonometric functions that depend only on $\phi_1$ or $\phi_2$ separately.

Now for physics: If a rocket is moving northward with rapidity $\phi_v$ relative to you, and it fires a projectile in the northward direction with rapidity $\phi_w$ (relative to the rocket), what is the rapidity of the projectile relative to you?