Question

How do you verify `tan^2(x) / sec(x) = sec(x) - cos(x)`?

Answer 1

First prove that `secx-cosx = (sinx)(tanx)`. Then, mess with the equation some to get the right answer.

Explanation:

(Lightly used source: https://answers.yahoo.com/question/index?qid=20080831084348AAj0BJV)

Start with the identity `sin^2x+cos^2x=1`

Subtract `cos^2x` from both sides:

`sin^2x = 1 - cos^2x`

Divide both sides by `cosx`.

`sin^2x/cosx = 1/cosx - cosx`

Now simplify. Remember sin/cos = tan, and 1/cos = sec

`sinx * sinx/cosx = secx - cosx`

`sinx*tanx = secx-cosx`

There. Now that we've proven that `(sinx)(tanx)=secx-cosx`, all we have to do is show that `tan^2x/secx = (sinx)(tanx)`.

`tan^2x/secx = tanx * tanx/secx = tanx*(tanx)(cosx) = tanx*sinx`

There we go! If this doesn't make sense, don't worry. Let me know in the 'comments' and I'll give you a list of the trigonometric identities I used to solve this.