tan(x + (pi/2)) = -cotx?

Question

Prove this identity and show all steps.

Answer 1

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

sin(x+pi/2)=sinx cos(pi/2) + sin(pi/2) cosx= cosx
cos(x+pi/2)=cosx cos(pi/2) - sin(pi/2)sinx= -sinx
Plug these two results to the above formula, you have

tan(x+pi/2)=cosx/(-sinx)= -cotgx

Answer 2

Yes, this is always true.

You can see this if you draw a set of x an y axes and then let the x-axix be the initial side of the angle and some other line from the origin i the 1st quadrant be the terminal side of the angle x.

Then rotate the terminal side 90 degrees to create an angle = x+90 degrees. There are a couple of congruent triangles where the y value in the second quadrant is = to the x value in the 1st quadrant and the x value in the second quadrant is = to - the y value in the first quadrant.

This is why the sin turns into the cos and the tan into the cot and the cos into the sin. I know this would be easier to explain if I could draw pictures but we don't have that capability here.