  In 2007, Tachiya gave necessary and sufficient conditions for the
transcendence of certain infinite products involving Fibonacci numbers $F_k$
and Lucas numbers $L_k$. In the present note, we explicitly evaluate two
classes of his algebraic examples. Special cases
are$$\prod_{n=1}^{\infty}(1+\frac{1}{F_{2^n+1}})=\frac{3}{\phi}, \qquad
\prod_{n=1}^{\infty}(1+\frac{1}{L_{2^n+1}})=3-\phi\, ,$$where
$\phi=(1+\sqrt{5})/2$ is the golden ratio.
