Equating trig identities

Mar 2017
358
3
Massachusetts
Hi,

I hope someone can help. It would be great if someone could explain how 2((secx)^4 + 2(secx)^2(tanx)^2) equals -2(cos(2x)-2)(secx)^4

I know that the expressions are related in some way, I just can't pinpoint what.

- Olivia
 

Prove It

MHF Helper
Aug 2008
12,897
5,001
It might be worth converting to sines and cosines...
 

skeeter

MHF Helper
Jun 2008
16,217
6,765
North Texas
how 2((secx)^4 + 2(secx)^2(tanx)^2) equals -2(cos(2x)-2)(secx)^4
working on the right side to get the left ...

$-2(\cos(2x)-2)\sec^4{x}$

$-2(2\cos^2{x}-1-2)\sec^4{x}$

$2(3-2\cos^2{x})\sec^4{x}$

$2(3\sec^4{x} - 2\sec^2{x})$

$2(\sec^4{x} + 2\sec^4{x} - 2\sec^2{x})$

$2[\sec^4{x} + 2\sec^2{x}( \sec^2{x} - 1)]$

$2(\sec^4{x} + 2\sec^2{x}\tan^2{x})$