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For a unit-speed space curve, setThese form the Frenet trihedron. With the signed torsion convention compatible with the question, the Frenet-Serret formulas areFor an oriented surface with unit normal $N$, the signed geodesic curvature is

On the unit sphere $N=\alpha$. Since $\alpha \cdot t=0$, differentiation givesWrite $\alpha =An+Bb$. Then $A=-1/\kappa$, and differentiating this expression and comparing the $n$ coefficient with $\dot{\alpha }=t$ givesThus, wherever the displayed quantities are defined,Finally,which is the stated curvature decomposition for a spherical curve.