Question

How do you verify the identity `(tanx+secx)(1-sinx)=cosx`?

Answer 1

See below.

Explanation:

Use the following identities to simplify the left-hand side:

•`tanbeta = sinbeta/cosbeta`

•`secbeta = 1/cosbeta`

Start the simplification process:

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

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

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

Now, rearrange the pythagorean identity `sin^2beta + cos^2beta = 1` for `cos^2beta` to get `cos^2beta = 1- sin^2beta`. Applying this to our problem:

`cos^2x/cosx = cosx`

`((cosx)(cosx))/cosx = cosx`

`(cancel(cosx)cosx)/cancel(cosx) = cosx`

`cosx = cosx`

Identity proved!!

Hopefully this helps!