Find $\lim_{x\rightarrow \frac{\pi}{4}}\left(\frac{\sin x}{\cos x}\right)^\left(\frac{\sin 2x}{\cos 2x}\right)$

Question

Find $$\lim_{x\rightarrow \frac{\pi}{4}}\left(\frac{\sin x}{\cos x}\right)^\left(\frac{\sin 2x}{\cos 2x}\right)$$

My work so far: $$\lim_{x\rightarrow \frac{\pi}{4}}\left(\frac{\sin x}{\cos x}\right)^\left(\frac{\sin 2x}{\cos 2x}\right)=\lim_{x\rightarrow \frac{\pi}{4}}\tan^{\tan 2x}x=1^{\infty}$$

Answer 1

HINT: write your term in the form $$e^{\lim_{x\to \frac{\pi}{4}}\frac{\ln(\tan(x)}{\cot(2x)}}$$ and use L'Hospital