Question

PLEASE VERIFY THE IDENTITY

cot(theta-pi/2) = -tan theta

Answer 1

Some identities:

cos(a-b) = cos(a)cos(b) + sin(a)sin(b)
sin(a-b) = sin(a)cos(b) - cos(a)sin(b)
cot(a) = 1/tan(a)

Therefore (substituting x for theta):

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

cos(pi/2) =0
sin(pi/2) =1

So: sin(x)/-cos(x) = -tan(x)