Verify the identity:
tan(x+pi/2)=-cotx

First we use sin(a+b)= sinacosb+sinbcosa
and cos(a+b)=cosa cosb -sinasinb

tan(x+pi/2)= sin(x+pi/2) / cos(x+pi/2)
and sin(x+pi/2) = sinxcospi/2 + sinpi/2cosx =cosx, 
cos(x+pi/2) = cosxcospi/2- sinxsinpi/2= - sinx, 
 because cospi/2 =0,  and sinpi/2=1

=tan(x+pi/2)= sin(x+pi/2) / cos(x+pi/2)= cosx / -sinx = -1/tanx = -cotx
from where tan(x+pi/2)=-cotx