Question

How do you prove `secx(csc^2x)-csc^2x = secx / (1 + cosx)`?

Answer 1

We have:

`csc^2x(secx- 1) = secx/(1 + cosx)`

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

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

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

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

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

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

As required.

Hopefully this helps!