Limit for Solving Trigonometric Equation near pi/2

[fermio]

Homework Statement

$$\lim_{x\to\frac{\pi}{2}}(x-\frac{\pi}{2})\tan x$$

Homework Equations

answer is (-1)

The Attempt at a Solution

$$\lim_{x\to\frac{\pi}{2}}(x-\frac{\pi}{2})\tan x= \lim_{x\to\frac{\pi}{2}}(x-\frac{\pi}{2})\cdot\frac{1-\cos(2x)}{\sin(2x)}$$

 

[morphism]

Try rewriting it as (x - pi/2)/(cotx). Why is this helpful?