How do you write `(sec theta - 1)(sec theta + 1)` in terms of sine and cosine?

Using the Identity : `sec^2theta=tan^2theta+1`, we have,

`(sectheta-1)(sectheta+1)=sec^2theta-1=sin^2theta/cos^2theta`.

Otherwise,

`(sectheta-1)(sectheta+1)`,

`=sec^2theta-1`,

`=1/cos^2theta-1`,

`=(1-cos^2theta)/cos^2theta`,

`=sin^2theta/cos^2theta`, as above!

First, you should multiply the expression and simplify as far as you can. Then, write everything in terms of sine and cosine.

Here are the identities we'll use:

`color(white){color(black)( (sectheta=1/costheta, qquadqquad(1.1)), (sec^2theta=1/cos^2theta, qquadqquad(1.2)), (tantheta=sintheta/costheta, qquadqquad(2.1)), (tan^2theta=sin^2theta/cos^2theta, qquadqquad(2.2)), (sin^2theta+cos^2theta=1, qquadqquad(3.1)), (sin^2theta/cos^2theta+cos^2theta/cos^2theta=1/cos^2theta, qquadqquad(3.2)), (tan^2theta+1=sec^2theta, qquadqquad(3.3)):}`

Some notes: identity `(1.2)` was achieved by squaring both sides of identity `(1.1)` (same with `(2.2)` and `(2.1)`).

Similarly, identity `(3.2)` was achieved by dividing all the terms in identity `(3.1)` by `cos^2theta`. Then, identity `(3.3)` was reached by simplifying identity `(3.2)` using previously-proved identities `(1.2)` and `(2.2)`

Now, here's the expression:

`color(white){color(black)( ((sectheta-1)(sectheta+1), qquadqquad"The problem"), (sec^2theta+sectheta-sectheta-1, qquadqquad"Multiplying out the expression"), (sec^2thetacolor(red)cancelcolor(black)(+sectheta-sectheta)-1, qquadqquad"Like terms cancel out"), (sec^2theta-1, qquadqquad"Rewrite the above step"), (tan^2theta+1-1, qquadqquad"Replace "sec^2theta" with "tan^2theta+1" using identity "(3.3)), (tan^2thetacolor(red)cancelcolor(black)(+1-1), qquadqquad"Like terms cancel out"), (tan^2theta, qquadqquad "Rewrite the above step"), (sin^2theta/cos^2theta, qquadqquad "Replace "tan^2theta" with "sin^2theta/cos^2theta" using identity "(2.2)):}`

That's the answer. Hope this helped!