Tan (x+pi/2) = negative cot x

I'm only going to alter the left hand side. The right side will stay the same the entire time

I'll use the identity tan (x) = sin (x) / cos (x) and cot (x) = cos (x) / sin (x)

I'll also use sin (x+y) = sin (x) cos (y) + cos (x) sin (y) and cos (x+y) = cos (x) cos (y) - sin (x) sin (y)

So with that in mind, this is how the steps would look:

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

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

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

(sin (x) * 0+cos (x) * 1) / (cos (x) * 0-sin (x) * 1) = - cot x

(0+cos (x)) / (-sin (x) - 0) = - cot x

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

-cot x = - cot x

Identity is confirmed