https://www.cut-the-knot.org/proofs/sine_cosine.shtml

a quadrilateral is inscribed in a circle having diameter 1.

The page lists some equivalences (down towards the bottom) between the angles alpha and beta and the length of the sides of the quadrilateral.

I understand from the Sine Rule that sin(alpha) = AC, and sin(beta) = DC.

What I do not understand is how the equivalences cos(alpha) = AB and cos(beta) = BD were derived. Did a step go missing there somewhere? I'm scratching my head trying to prove it.

Thanks,
Matt

Perhaps the apppeal to the Sine Law is misleading. Sorry for that. The simple truth is that both sine and cosine are originally defined in a right triangle as, respectively, the ratio of the opposite leg to the hypotenuse and the ratio of the adjacent leg to the hypotenuse. In triangle ABC with the hypotenuse BC (since it's a diameter, angle A is right), sin(α) = AC/BC and cos(α) = AB/BC. Now, because, by our choice BC = 1, sin(α) = AC and cos(α) = AB.