Question

How do you prove `Sec(x) - cos(x) = sin(x) * tan(x)`?

Answer 1

Recall that `sectheta = 1/costheta` and that `tantheta = sintheta/costheta`

`1/cosx - cosx = sinx xx sinx/cosx`

Now put the left side on a common denominator:

`1/cosx - (cosx xx cosx)/cosx = sinx xx sinx/cosx`

Simplify:

`(1 - cos^2x)/cosx = sin^2x/cosx`

Now use the modified pythagorean identity `sin^2x + cos^2x = 1`: `sin^2theta = 1 - cos^2theta`.

`(sin^2x)/cosx = sin^2x/cosx ->` Identity proved!!

Practice exercises:

1. Prove the following identities, using the quotient, pythagorean and reciprocal identities.

a) `1/tanx + tanx = 1/(sinxcosx)`

b) `cos^2x = (cscxcosx)/(tanx + cotx)`

Hopefully this helps, and good luck!