# How do we prove that $\lim_{x\to\pi/4}\cos(2x)/(1-\sqrt{2}\sin(x)) = 2$ without L'Hôpital? [closed]

How can I prove the following limit equals two without LHopital?

Through L'Hôpital, I reached this conclusion:

$$\lim _{x \rightarrow \frac{\pi}{4}} \frac{\cos 2 x}{1-\sqrt{2} \sin x}=2$$

\begin{align*} \cos(2x) = \cos^{2}(x) - \sin^{2}(x) = 1 - 2\sin^{2}(x) = (1 - \sqrt{2}\sin(x))(1+\sqrt{2}\sin(x)) \end{align*}