Entry $(1,1)$ of $C$ equals $R_0(r) + i\,\Theta_0(\theta)$

This is the first of the four coefficient checks behind the separation claim $C^{(\varsigma)}(r,\theta) = R^{(\varsigma)}(r) + i\,\Theta^{(\varsigma)}(\theta)$ of Hintz (text after eq. (7)). With $C^{(\varsigma)} = (\det X)X^{-1}(c^{(\varsigma)}I - B^{(\varsigma)})(X^*)^{-1}$ defined from the data of eqs. (4)–(5), and with the independent explicit formulas

($K = -(r^2+a^2)\omega + am$, $\Delta' = 2(r-M)$), the theorem states that at every point of $U$ the $(1,1)$ entry of $C^{(\varsigma)}$ equals $R_0^{(\varsigma)}(r) + i\,\Theta_0^{(\varsigma)}(\theta)$. The proof multiplies by the constant $\det X = Q^2-M^2$, expands the triple product $\operatorname{adj}(X)(cI-B)\operatorname{adj}(X)^*$ entrywise, clears the denominators $\bar\rho,\bar\rho^*,\Delta,\sqrt\Delta,\sin\theta,Q,M^2-Q^2$ (all nonzero on $U$), and reduces powers of $\sqrt\Delta$, $\cos\theta$ and $i$ with $\sqrt\Delta^2=\Delta$, $\cos^2\theta = 1-\sin^2\theta$, $i^2=-1$; the two sides then agree as polynomial identities.