Question

verify the identity cos(pi - theta) + sin(pi/2 + theta)=0

Answer 1

cot ( theta - pi / 2 )= cos ( theta - pi / 2 ) / sin ( theta - pi / 2 )

cos ( theta - pi / 2 )= cos(theta) cos (pi/2) + sin(theta) sin (pi/2)
= cos(theta) (0) + sin(theta) (1)
= sin(theta)

sin ( theta - pi / 2 )= sin(theta) cos(pi/2) - cos(theta) sin(pi/2)
= sin(theta) (0) - cos(theta) (1)
= -cos(theta)
so you get,
cot ( theta - pi / 2 )= sin(theta) / -cos(theta)
cot ( theta - pi / 2 )= -tan(theta)

Answer 2

use these identities
cos(pi - x) = - cos(x)
sin(pi/2 + x) = cos(x)