Question

How do you simplify `(sec y - tan y) (sec y + tan y) / sec y`?

Answer 1

Apply the identity `sectheta = 1/costheta` and `tantheta = sintheta/costheta`.

`1/cosy - siny/cosy((1/cosy + siny/cosy)/(1/cosy))`

`=((1 - siny)(1 + siny)/cosy)/(cancel(cosy) xx 1/cancel(cosy))`

`= (1 - sin^2y)/cosy`

Apply the pythagorean identity `cos^2theta = 1 - sin^2theta`

`= cos^2y/cosy`

`=cosy`

Hopefully this helps!

Answer 2

`cosy`

Explanation:

Given expression
`=((secy-tany)(secy+tany))/secy`

`=(sec^2y-tan^2y)*cosy`
`=1*cosy=cosy`
[Using formula `(sec^2y-tan^2y)=1 and 1/secy=cosy`]