How do you verify this Identity? Cot (x - pi / 2) = - tan x

Answer 1

You can verify by using trig identity

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

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

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

= sin x / -cos x

= -tan x

Answer 2

The easiest way is to draw a picture, but this may not be an acceptable answer, although you can prove a "picture" geometrically. I am not sure how to get it analytically.

This statement is the equivalent of the analytic geometry statement that the slope of a line perpendicular to a given line is the negative reciprocal of the given line.

The figure you need is a "V' (not necessarily symmetric) with the vertex at the origin.

Lay off equal line segments on each arm of the "v".

Then prove that the acute triangle to the right of the origin and the acute triangle with base on the y axis are congruent. The look at the tangents of the angles in the picture.

If you can't see it with this information, please let me know and I will try to send you a picture.