Proving $\pi$ irrational: help with Lambert's proof. "Circularity"?

Question

This expression is irrational. $$\tan(x)=\frac{x}{1-\frac{x^2}{3-\frac{x^2}{5-...}}}$$ But then he used the fact that $\tan{\frac{\pi}{4}}=1$, so $\frac{\pi}4$ is irrational. But how can we use tangent function here if we are proving the irrationality of $\pi$.
And is there any simpler proof also? (Using elementary functions and operations)

Answer 1

Your first sentence is false. Probably the correct version is that your continued fraction is irrational if $x$ is rational. Therefore, since we know that $\tan(\frac{π}{4})$ is rational, $\frac{π}{4}$ cannot be rational.