A generalization of the Pythagorean trigonometric identity

"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.

An interesting discussion about this generalization is available at MathSE.

For more implications of $(1)$ I invite you to see The theoretical importance of the half-angle formulas.