Let Tλ(x)=cos(λarccosx), −1≤x≤1, where λ>1 is not an integer.
For a certain set of λ's which are irrational, the density of the unique absolutely continuous measure invariant under Tλ is determined exactly. This
is accomplished by showing that Tλ is differentially conjugate to a piecewise linear Markov map whose unique invariant density can be computed
as the unique left eigenvector of a matrix.