Verify the identity. cot{x- pi/2}=-tan x

Question

I'm just not sure how to go about doing this.  Thank you for your help.

Answer 1

cot(x - π/2) = 1 / tan(x - π/2) = 1 / [sin(x - π/2) / cos(x - π/2)]

 

= cos(x - π/2) / sin(x - π/2) = cos(π/2 - x) / [-sin(π/2 - x)]

 

= sin(x) / [-cos(x)] = -[sin(x) / cos(x)] = -tan(x)

 

 

We used the properties:

 

cos(-u) = cos(u)

sin(-u) = -sin(u)

cos(π/2 - u) = sin(u)

sin(π/2 - u) = cos(u)