= Solution
For both substrates, the <alkene> pi bond attacks <bromine> and polarizes/breaks Br–Br. A <bromine> <lone pair> completes a three-membered <bromonium ion>, and bromide subsequently attacks a <carbon> from the opposite face, cleaving that C–Br bridge bond. \b[The two new C–Br bonds are anti.] A free carbocation that rotates before capture is not needed.
In trans-fused $A$, the fixed <conformation> makes the alternative anti-opening pathways diastereomeric. A chair-like, trans-diaxial opening is favored over a geometrically strained alternative that requires a twist/boat-like arrangement of the six-membered ring. The <Fürst-Plattner rule> describes this conformational preference. \b[One depicted <enantiomer> of $A$ therefore gives essentially one dibromide <diastereomer>], with the two bromines occupying the trans-diaxial arrangement in its favored chair. If a racemic sample of $A$ were used, its other <enantiomer> would of course give the mirror product as well.
For cis-fused $B$, the accessible mirror-related folds and the equivalent anti pathways produce mirror-related dibromides with equal rates in an <achiral> environment. Adding the two bromines destroys the symmetry that made the starting <configuration> <achiral>. \b[$B$ gives a racemic dibromide pair.] The racemate follows from equivalent <enantiomer>-forming pathways, not from a loss of anti stereospecificity.
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