= Solution
{c}
By definition $v/c=\tanh\theta$ and $v'/c=\tanh\theta'$. Using the addition identity for the <hyperbolic tangent> and the <rapidity> sum from (i),
$$
\frac{v''}c=\tanh(\theta+\theta')=\frac{v/c+v'/c}{1+vv'/c^2}.
$$
Thus the <velocity-addition formula> is
$$
\boxed{v''=\frac{v+v'}{1+vv'/c^2}=\frac{c^2(v+v')}{c^2+vv'}.}
$$
The denominator is positive for the permitted subluminal signed velocities. For the nonnegative speeds in (iii), the composite is also nonnegative and remains below $c$.
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