= Solution
{c}
In the common $ct,x_1$ plane, multiplication of the two <Lorentz boost> matrices gives
$$
\begin{pmatrix}\cosh\theta&\sinh\theta\\\sinh\theta&\cosh\theta\end{pmatrix}
\begin{pmatrix}\cosh\theta'&\sinh\theta'\\\sinh\theta'&\cosh\theta'\end{pmatrix}
=\begin{pmatrix}\cosh(\theta+\theta')&\sinh(\theta+\theta')\\\sinh(\theta+\theta')&\cosh(\theta+\theta')\end{pmatrix}.
$$
This follows from the addition identities for <hyperbolic sine> and <hyperbolic cosine>. The transverse coordinates are unchanged by both transformations. Consequently the <rapidity> of $\overline O$ relative to $O$ is
$$
\boxed{\theta''=\theta+\theta'.}
$$
The signed rapidities add because both boosts use the same spatial axis and the stated matrix convention.
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