Question

How do you show that `cosx/(1 - sinx) = secx+ tanx`?

Answer 1

We know that `sectheta = 1/costheta` and `tantheta= sintheta/costheta`.

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

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

Multiply the left side by the conjugate of the denominator. The conjugate of `a + b` is `a- b`, for example.

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

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

Use the identity `sin^2theta + cos^2theta= 1-> cos^2theta = 1- sin^2theta`.

`(cosx(1 + sinx))/cos^2x = (1 + sinx)/cosx`

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

`LHS = RHS`

Identity Proved!

Hopefully this helps!