Verify the identity. tan x plus pi divided by two = -cot x

Hello! Probably you mean "tangent of (x plus pi / 2)", not "(tangent of x) plus pi / 2", i.e. tan(x + pi / 2) = - cot (pi / 2).
It is correct. The simplest way to verify it is to recall that tan u = (sin u) / (cos u) and apply the known identities about the sine and cosine of a sum,
sin(u + v) = sin u cos v + cos u sin v and cos(u + v) = cos u cos v - sin u sin v.
Form these identities we obtain
tan(x + pi / 2) = (sin(x + pi / 2)) /(cos(x + pi / 2)) =(sin x cos(pi / 2) + cos x sin(pi / 2)) /(cosx cos(pi / 2) - sin x sin(pi / 2)).
Recall that sin( pi/2) = 1 and cos( pi/2) = 0, and the equality becomes
tan(x + pi / 2) = (cos x ) /( - sin x ),  which is indeed equal to -cot x.
 
Actually, one can directly use the identities sin(x+pi/2)=cos x and cos(x+pi/2)=-sin x.