Question

Verify the identity cot{0-pi/2=-tan 0

Answer 1

use the identities:
tan(x)=sin(x)/cos(x)
cot(x)=cos(x)/sin(x)
and
sin(π/2)=1, cos(π/2)=0
sin(0)=0, cos(0)=1

Answer 2

I think she means θ when using 0
I'll use x to avoid copy/paste of special characters.

one way:
recall the definition of co-functions: function of complementary angle. So,

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

so, cot(x-pi/2) = -cot(pi/2 - x) = -tan(x)

or, use cot addition formula:

cot(x-pi/2) = (1+cotx cot pi/2)/(cot pi/2 - cot x) = (1+0)/(0-cotx) = 1/-cotx = -tanx

Answer 3

Thanks Steve, good thought!