# A spherical version of the generalized half-angle formulas

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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