How can tan 4x be simplified or sec 2x?

1 Answer
Aug 7, 2015

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))#