The following is a generalization of the half-angle formulas presented at Nabla - Applications of Trigonometry.
Generalization. Let $a$, $b$, $c$, $d$ be the sides of a general convex quadrilateral, $s$ is the semiperimeter, and $\alpha$ and $\gamma$ are opposite angles, then $$ad\sin^2{\frac{\alpha}{2}}+bc\cos^2{\frac{\gamma}{2}}=(s-a)(s-d)\qquad\qquad bc\sin^2{\frac{\gamma}{2}}+ad\cos^2{\frac{\alpha}{2}}=(s-b)(s-c).\tag{1}$$ I am interested in the following: a) If possible, a spherical generalization of $(1)$. b) If a) is answered in the affirmative, what classical metric relations of spherical trigonometry would follows from a) (or a particular case of a)?
This question is related to a previous question.
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