How do you prove #tan^4 x + 2tan^2 x + 1 = sec^4 x#?

Question

15839 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-06-12T16:26:58

see below

we need the identity

#color(red)(1+tan^2x=sec^2x)#

#LHSrarrtan^4x+2tan^2x+1#

#=tan^4x+tan^2x+color(red)(tan^2x+1)#

#=tan^4x+tan^2x+sec^2x#

#=tan^2xcolor(red)((tan^2x+1))+sec^2x#

#=tan^2x(sec^2x)+sec^2x#

#=sec^2xcolor(red)((tan^2x+1))#

#=sec^2xsec^2x#

#=sec^4x#