"Discovery is seeing what everybody else has seen, and thinking what nobody else has thought."
-- Albert Szent-Györgyi (1893 - 1986)
I don't know how I missed it, but the Pythagorean trigonometric identity is a special case of the generalized half-angle formulas.
The following is a generalization of the half-angle formulas presented at Nabla - Applications of Trigonometry for a triangle.
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}
For a proof of (1) see pp. 8 in MATINF.
The Pythagorean identity as a special case
In (1), consider the case when a=c, b=d and \alpha=\gamma so that the quadrilateral is a parallelogram . Then
ab\sin^2{\frac{\alpha}{2}}+ab\cos^2{\frac{\alpha}{2}}=\frac{-a+b+c+d}{2}\cdot{\frac{a+b+c-d}{2}}=ab.Dividing both sides by ab you get
\sin^2{\frac{\alpha}{2}}+\cos^2{\frac{\alpha}{2}}=1.
Or by making \frac{\alpha}{2}=\theta,
\sin^2{\theta}+\cos^2{\theta}=1,
which is the Pythagorean trigonometric identity.
For more implications of (1) I invite you to see The theoretical importance of the half-angle formulas.
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