Question

How would you prove or disprove `cotx - cosx/cotx = cos^2x/(1 + sinx)`?

Answer 1

Rewrite all terms in `cotx` as `cosx/sinx`.

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

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

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

Rewrite the right-hand side using the identity `sin^2x + cos^2x = 1`.

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

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

`cosx/sinx - sinx = 1 - sinx`

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

The identity is false, because no matter what you do with the left hand side, you will never be able to get on the right.

Hopefully this helps!