= Solution
The Mayer--Vietoris sequence for the splice identifies the companion meridian with the pattern longitude and kills exactly the relation already killed when passing from $X_{U,P}$ to $X_P$. It therefore gives a canonical isomorphism
$$
\phi:H_1(X_P)\xrightarrow{\ \cong\ }H_1(X_{CP}).
$$
Apply multiplicativity of <Turaev torsion> to the torus union defining $X_{CP}$, and use part (b) to replace the torsion of $X_{U,P}$. The factors $1-[\lambda_U]$ cancel against the peripheral factor from the companion exterior. Using
$$
\tau(X_K)=\frac{\Delta_K}{1-[\mu_K]}
$$
on the three knot exteriors leaves
$$
\boxed{\Delta_{K_CK_P}
=\iota_*^C(\Delta_{K_C})\,\phi(\Delta_{K_P}).}
$$
Under a one-variable identification, $\iota_*^C$ sends the companion variable to $t^w$, where $w=[U]\in H_1(X_P)\cong\mathbb Z$ is the <winding number of a satellite pattern>. Thus this is also the usual <Satellite formula for the Alexander polynomial>.
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