To verify tan (x-3pi/2) = -cotx.
Consider the left side:
Tan (x-3pi/2) = Tan (x-3pi +2pi) as tan function is 2pi periodic.
= - tan (supplementary angle of x+Pi/2)
= -tan (pi/2-x)
=-cotx = RHS, as in triangle ABC with B as triangle, and A as x+pi/2, and C as pi/2 - x, tanA = cotB. Or Tan (x+pi) = cot(pi-x).