Question

How do you verify the identity `cosxcotx = cscx - sinx`?

Answer 1

All the identities you will need:
`cotx= cosx/sinx`
`cos^2x= 1-sin^2x`
`1/sinx= cscx`

Starting:
`cosxcotx= cscx-sinx`

Apply number 1 on the list:

`cosx*cosx/sinx= cscx-sinx`

Simplify:

`cos^2x/sinx= cscx-sinx`

Apply number 2 on the list:

`(1-sin^2x)/sinx= cscx-sinx`

Split the numerator:

`1/sinx-sin^2x/sinx= cscx-sinx`

Apply number 3 on the list:

`cscx-sinx=cscx-sinx`

Answer 2

Kindly go through a Proof in the Explanation.

Explanation:

We have, `cosxcotx+sinx`,

`=cosx*cosx/sinx+sinx`,

`=(cos^2x+sin^2x)/sinx`,

`=1/sinx`,

`:. cosxcotx+sinx=cscx`.

`rArr cosxcotx=cscx-sinx`, as desired!