Question

How do you verify the identity `tanthetacsc^2theta-tantheta=cottheta`?

Answer 1

Proof below

Explanation:

`tantheta * csc^2theta - tantheta`
`=sintheta/costheta * (1/sintheta)^2 - sintheta/costheta`
`=sintheta/costheta * 1/sin^2theta - sintheta/costheta`
`=1/(sinthetacostheta) - sintheta/costheta`
`=(1-sin^2theta)/(sinthetacostheta)`
`=cos^2theta/(sinthetacostheta)`
`=costheta/sintheta`
`=cottheta`

Note that `sin^2theta + cos^2theta = 1`, therefore `cos^2theta = 1- sin^2theta`

Answer 2

`LHS=tantheta * csc^2theta - tantheta`

`=tantheta ( csc^2theta - 1)`

`=tantheta ( 1+cot^2theta - 1)`

`=tantheta * cot^2theta`

`=cottheta=RHS`