I've already solved this porblem using L'Hospital rule, and now I'm required to solve it without using this rule, rather using the given condition.

Find:

Lim (sinx)^tanx

x-->pi/2

Given that: Lim (1+1/x)^x= e

x-->infinity

The answer is, as you might have guessed, 1. But, can anyone do it absolutely without using L'Hopital rule? I tried to do, but it was only elegant solution, so that's worthless.