Approximating $\frac{t^2}{3-\frac{t^2}{5-\frac{t^2}{7-\frac{t^2}{9-\cdots}}}}$ What is a good approximation for $$\omega=\frac{t^2}{3-\frac{t^2}{5-\frac{t^2}{7-\frac{t^2}{9-\cdots}}}}$$ This will be used to find $$T=\frac{t}{1-\omega}$$ Without using Lambert's continued fraction

From Lambert's continued fraction we have that:
$$\tan z = \frac{z}{1-\frac{z^2}{3-\frac{z^2}{5-\frac{z^2}{7-\frac{z^2}{\ldots}}}}}$$
holds for any $z$ in a neighboorhood of zero, hence, for small $t$'s:
$$\omega = 1-\frac{t}{\tan t}.$$
