The following is a generalization of the half-angle formulas presented in the following link for a triangle: http://www.nabla.hr/GE-AppTrigonomB1.htm

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 $$ad\sin^2{\frac{\alpha}{2}}+bc\cos^2{\frac{\gamma}{2}}=(s-a)(s-d)\tag{1}$$

and

$$bc\sin^2{\frac{\gamma}{2}}+ad\cos^2{\frac{\alpha}{2}}=(s-b)(s-c).\tag{2}$$

Bretschneider's Formula can be derived from $(1)$ and $(2)$ (see https://geometriadominicana.blogspot.com/search?updated-max=2020-11-21T06:09:00-08:00&max-results=7). I am surprised that this generalization seems to be unknown. This is my question: can these formulas be extended to spherical or hyperbolic geometry as suggested by work by G.A. Bajgonakova and A. Mednykh for Bretschneider's Formula? See for example:

https://www.researchgate.net/publication/265636400_On_Bretschneider's_formula_for_a_spherical_quadrilateral

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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