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

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Answer 1 · 2018-06-12T16:26:58

see below

we need the identity

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

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

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

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

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

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

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

$=sec^2xsec^2x$

$=sec^4x$

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