How do you prove that (1 - cos2x)/sin2x = tanx?

You need to know some identities for sin2x and cos2x:

sin2x = 2sinxcosx
cos2x = 1 - 2((sinx)^2); The "^" means raise to the power; "^2" means "raise to the power 2" or "square".


(1 - cos2x)/sin2x

= (1 - [ 1 - 2((sinx)^2) ] ) / (2sinxcosx)

= 2((sinx)^2) / (2sinxcosx)

= 2sinxsinx / 2sinxcosx

= sinx / cosx

= tan x.