Infinite products associated with the ±1 Thue-Morse sequence whose value is rational or algebraic irrational have been studied by several authors. In this short note we prove three new infinite product identities involving π, √2, and the ±1 Thue-Morse sequence, building on a result by Allouche, Riasat, and Shallit. We then use our method to find a new expression for a product appearing within the context of the Flajolet-Martin constant.