sin2x/sinx - cos2x/cosx = secx

Remember these important identities:

sin2x = 2sinxcosx
cos2x = cos^2x - sin^x
= 2 cos^x - 1
= 1-2sin^2x.

That second identity for cos2x looks promising, since you will be dividing by cosx. Try it, and see if you make progress.