Ans: -1
Question says: \( \lim\limits_{x \to \pi/2} (x-\pi/2)Tanx\)
Now, lets assume Cotx = t
So, question changes to : \( \lim\limits_{t\to 0} (arc cot(t)-\pi/2)/t\) which if a 0/0 indeterminate fom and thus you can apply L’hopital rule as:
\(\lim\limits_{t\to 0} (arc cot(t)-\pi/2)/t = \lim\limits_{t\to 0} -1/(1+t^2) = -1\)