Evaluation of Tachiya's algebraic infinite products involving Fibonacci and Lucas numbers

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$$\prod _{n=1}^{\mathrm{\infty }}(1+\frac{1}{F_{2^n+1}})=\frac{3}{\varphi },\prod _{n=1}^{\mathrm{\infty }}(1+\frac{1}{L_{2^n+1}})=3-\varphi ,$$where $\varphi =(1+\sqrt{5})\mathrm{/}2$ is the golden ratio.