Idea for $\lim\limits_{x \to \frac{\pi}{2}} \left( \tan \left( \tfrac{\pi}{4} \sin x\right)\right)^{1/ ( \tan (\pi \sin x))}$ $$\lim_{x \to \frac{\pi}{2}} \tan \left(\frac{\pi}{4}\sin x\right)^\left({\dfrac 1{\tan(\pi \sin x)}}\right).$$
Need help for this. I tried using trigonometric identities but nothing seems to fit. I'm tired and clueless.
ps. can't use lhopital

Alternatively:
$$\lim_{x \to \frac{\pi}{2}} \tan \left(\frac{\pi}{4}\sin x\right)^\left({\dfrac 1{\tan(\pi \sin x)}}\right)=\\
\lim_{x \to \frac{\pi}{2}} \left[1+\tan \left(\frac{\pi}{4}\sin x\right)-1\right]^\left({\dfrac 1{\tan(\pi \sin x)}}\right)=\\
\exp \left[\lim_{x \to \frac{\pi}{2}} \frac{\tan \left(\frac{\pi}{4}\sin x\right)-1}{\tan(\pi \sin x)} \right]=\\
\exp \left[\lim_{x \to \frac{\pi}{2}} \frac{\sin \left(\frac{\pi}{4}\sin x\right)-\cos \left(\frac{\pi}{4}\sin x\right)}{\sin(\pi \sin x)}\cdot \frac{\cos (\pi \sin x)}{\cos \left(\frac{\pi}{4}\sin x\right)}\right] =\\
\exp \left[\lim_{x \to \frac{\pi}{2}} \frac{\cos \left(\frac{\pi}{2}\sin x\right)}{\sin(\pi \sin x)}\right] =\\
\exp \left[\lim_{x \to \frac{\pi}{2}} \frac{1}{2\sin(\frac{\pi}{2} \sin x)}\right] =\exp \left[\frac12\right]=\sqrt{e}.
$$
