I have a problem that I have worked on for a long time and cannot solve. It is really bugging me. The directions are to show the steps in varifying these identities...

(cosx +sinx)/(cosx - sinx) - (cosx - sinx)/(cosx + sinx) =2tan(2x)

I keep getting sooo close but just can't seem to arrive at 2tan(2x)! AHHHHH!! Please help as I am in need for a brain rest. Thank you kindly.

Did you just cross multiply and get:

\frac{4cos(x)sin(x)}{cos^{2}(x)-sin^{2}(x)}?

If so, this is equal to 2tan(2x).

Because 4cos(x)sin(x)=2sin(2x)

Also, cos^{2}(x)-sin^{2}(x)=cos(2x)

Then, we get:

\frac{2sin(2x)}{cos(2x)}=2tan(2x)