Reversed-period identity for a purely periodic continued fraction

Let $\theta =[\overline{a_0,\dots ,a_n}]$, and let ${\theta }^{\prime }$ be its algebraic conjugate, the other fixed point of the Möbius transformation represented by $M(a_0)\cdots M(a_n)$. ThenThe identity follows by transposing the continued-fraction matrix and substituting $X=-1/Y$ in its fixed-point equation.