Question

Verify this identity: show work

cot(x-pi/2)= -tanx

Answer 1

`-tan(x) = cot(x - (\pi)/(2))`

`-tan(x) = (cos(x - (\pi)/(2)))/(sin(x - (\pi)/(2)))`

`-tan(x) = (sin(x))/(-cos(x))`

`-tan(x) = -tan(x)`

There :).

The most important thing to note here is that
`cos(x-(\pi)/(2)) = sin(x)`
and
`sin(x-(\pi)/(2)) = -cos(x)`

Normally, over time you end up memorizing these two identities since you use them so often, but proving them is very easy:


`cos(x-(\pi)/(2)) = cos(x)cos((\pi)/(2)) + sin(x)sin((\pi)/(2)) = cos(x)*0 + sin(x) * 1`

`= sin(x)`


`sin(x-(\pi)/(2)) = sin(x)cos((\pi)/(2)) - cos(x)sin((\pi)/(2)) = sin(x) * 0 - cos(x) * 1`

`= -cos(x)`