calculus - Not using Taylor series or Maclaurin series, prove $frac{1}{sqrt{1-x^2}}=1+frac{1cdot3}{2cdot4}x^2+frac{1cdot3cdot5}{2cdot4cdot6}x^4+cdots$

I’m trying to find a proof of
$$\frac{1}{\sqrt{1-x^2}} = 1+\frac{1\cdot3}{2\cdot4}x^2+\frac{1\cdot3\cdot5}{2\cdot4\cdot6}x^4+\cdots,$$
which doesn’t need Taylor or Maclaurin series, like the proof of Mercator series and Leibniz series.
I tried to prove it by using calculus, but I couldn’t hit upon a good proof.