Trigonometric simplification for distance on a sphere

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These are two equivalent equations for distance on a sphere:

hav(x)=sin(x/2)^2

dist1(lat1, lat2, dLon)=2*asin(hav(lat1-lat2)+cos(lat1)*cos(lat2)*hav(dLon))

dist2(lat1, lat2, dLon)=acos(sin(lat1)*sin(lat2)+cos(lat1)*cos(lat2)*cos(dLon))

(According to http://www.movable-type.co.uk/scripts... ) Indeed numerically they evaluate equal.

But why their difference does not simplify to 0?

diff(x,y,z)=dist1(x,y,z)-dist2(x,y,z)

diff.simplify_full()

Am I doing something wrong? Is there a way to have Sage prove the two forms are equivalent, or to transform one to another?

Thanks!