Question

Verify this identity
cot ( x - pi/2) = -tanx

Answer 1

`cot(x- ( \pi )/(2) )= (1)/(tan(x- ( \pi )/(2) ))`


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


`(sinx*cos( ( \pi )/(2))-cosx*sin( ( \pi )/(2)))/(cosx*cos( ( \pi )/(2))+sinx*sin( ( \pi )/(2))) = (-cosx)/(sinx)=-cotx=-1/tanx`

(cos π/2 = 0, sin π/2 = 1)

so


`cot(x- ( \pi )/(2) )= (1)/(tan(x- ( \pi )/(2) ))= (1)/(-1/tanx)=-tanx`