Limit as x approaches pi/2 of cos (x) / x - (pi/2)

plug in x=0 gives 0/0. This is one of the indeterminate forms (some others are ∞/∞ and 0·∞) use L'Hospital's Rule.

Differentiate the top and bottom of the fraction:

lim (cos x) / (x - π/2) = > lim (-sin x) / (1 - 0)

Now you can substitute x = π/2 without problem

lim = - 1/1 = - 1