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

I'm totally stuck.

tan(x + pi/2) = sin (x + pi/2) / cos(x + pi/2) = (*)

sin (x + pi/2) = sin x cos pi/2 + cos x sin pi/2 = cos x

cos(x + pi/2) = cos x cos pi/2 - sin x sin pi/2 = - sin x

(*) = (cos x)/(- sin x) = - cot x

alternative proof
tan (x + y) = (tan x + tan y)/(1 - tan x tan y) = tan y ( (tan x)/(tan y) +1) / [(tan y)( 1/(tan y) - tan x)] =

= [(tan x)/(tan y) +1]/[1(/tan y) - tan x]

as y->pi/2, tan y -> infinity, 1/tan y -> 0

therefore

tan (x + pi/2) = 1/(- tan x) = - 1/tan x = - cot x