Is this use of product notation legal?

Is this use of product notation legal? References So I\'m trying to prove something, I came across a sub-question that lead me to another question I thought of. Is the following legal? $$\\prod_{k=1}^{n} \\left(1-\\frac{1}{2k}\\right)/ \\prod_{k=1}^{n} \\left(1-\\frac{1}{2k+1}\\right) = \\prod_{k=1}^{n} \\left(\\frac{1-\\frac{1}{2k}}{1-\\frac{1}{2k+1}}\\right) = \\prod_{k=1}^{n} \\left(1-\\frac{1}{4k^2}\\right)$$ (Which is the answer at the back) EDIT: Sorry guys, I mean to say - If this is true? $$\\prod_{k=1}^{n} \\left(a_{k}\\right)/ \\prod_{k=1}^{n} \\left(b_{k}\\right) = \\prod_{k=1}^{n} \\left(\\frac{a_{k}}{b_{k}}\\right) $$ If so, why? It works out for the proof I am doing, but I cannot see mathematically why? Anyone care to explain? Thank you. Sorry the the small code, I don\'t know why it turned out this way.