  We prove the recent conjectures of Adams-Vogan and D. Prasad on the behavior
of the local Langlands correspondence with respect to taking the contragredient
of a representation. The proof holds for tempered representations of
quasi-split real K-groups and quasi-split p-adic classical groups (in the sense
of Arthur). We also prove a formula for the behavior of the local Langlands
correspondence for these groups with respect to changes of the Whittaker data.
