Question

Verify the identity.

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

Answer 1

work the left side

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

Now you should know that sin(x+pi/2) = cos(x) and cos(x+pi/2) = - sin (x)

if you do not know that you can obtain from the sin (x+y) and cos (x+y) formulaes doing y = pi/2.

Continue:

sin (x+pi/2) / cos (x+pi/2) = cos(x) / [- sin(x)] = - 1 / tan(x) = - cot (x).

Then, we have obtained the right side from the left side, and the identity is verified.