\(\lim\limits_{x \to \frac \pi2} (cosec \, x)^{tan \, x}\) \((1^\infty\, type)\)
\(= Exp \left\{\lim\limits_{x\to \frac\pi2} (cosec\, x-1) tan\,x\right\}\)
\(= Exp \left\{\lim\limits_{x\to \frac \pi2} \frac {1 - sin \,x}{cos \,x}\right\}\) \(\left(\frac 00 - case\right)\)
\(= Exp \left\{\lim\limits_{x\to \frac \pi2} \frac {-cos \,x}{-sin\,x}\right\}\) (By using D.L.H. Rule)
\(= Exp \left\{\lim\limits_{x\to \frac \pi2} cot \, x\right\}\)
\(= e^0 = 1\) \((\because cot \frac \pi 2 = 0)\)
Hence, \(\lim\limits_{x\to \frac \pi 2} (cosec \, x)^{tan\, x} = 1\).
