Let's say we looked at #tan4x#. We can use the following identities:
#tan4x = (sin4x)/(cos4x)#
#sin2x = 2sinxcosx#
#cos2x = cos^2x - sin^2x#
#=> (2sin2xcos2x)/(cos^2 2x - sin^2 2x)#
#= (4sinxcosx(cos^2x - sin^2x))/((cos^2x - sin^2x)^2 - (2sinxcosx)^2)#
#= (4sinxcosx(cos^2x - sin^2x))/((cos^2x - sin^2x)^2 - 4sin^2xcos^2x)#
#= color(blue)((4sinxcosx(1 - 2sin^2x))/((1 - 2sin^2x)^2 - 4sin^2xcos^2x))#
I don't know if you can get it any simpler; it's all #sinx# and #cosx# now, though.
You could also have used:
#tan(2x+2x) = (tan(2x)+tan(2x))/(1-tan(2x)tan(2x))#
#= (2tan(2x))/(1-tan^2(2x))#
but that's gonna be uglier to simplify (unless you stop here).
#sec(2x)# is much simpler.
#= 1/(cos(2x)) = 1/(cos^2x - sin^2x) = color(blue)(1/(1-2sin^2x))#
