Question

How do I prove this identity: cos^2x - cos^4x = cos^2xsin^2x? o.o

Answer 1

well for one you can factor out cos^2x and see that you'll arrive with an identity

Answer 2

maybe start with
\[\cos^2(x)-\cos^4(x)=\cos^2(x)(1-\cos^2(x))\]

Answer 3

what do you mean?

Answer 4

let \(\cos^2 x = a\)

\[\cos^2 x - \cos^4 x = a - a^2\]
agree?

Answer 5

yes

Answer 6

so let's factor out a \[a(1 - a)\] right?

Answer 7

yup

Answer 8

let's substitute a back to \(\cos^2 x\)

\[\cos^2x(1 - \cos^2 x)\]

Answer 9

ohhh and then cos2x(1-sin2x)?

Answer 10

oh no. boo

Answer 11

uhmm no.... \(1 - \cos^2 x \ne 1 - \sin^2x\) however....let the identity \(\sin^2 x + \cos^2 x = 1\) be your guide

Answer 12

yup i was looking at that

Answer 13

so then what happens after cos2x(1−cos2x)

Answer 14

try to make \("\sin^2 x + \cos^2 x = 1"\) look like \("1 - \cos^2 x"\) can you d that??

Answer 15

isn't that for sin2x?

Answer 16

yup!

Answer 17

what do I do with it? o.o

Answer 18

\[\cos^2 x (1 - \cos^2x) = \cos^2 x (\sin^2 x)\]

Answer 19

you have just proven it :D

Answer 20

ohhhhhhh. omg, i need to look at the right side too >< thank you!

Answer 21

i keep forgetting

Answer 22

hahaha but remember that in proving you can only touch ONE side..

Answer 23

but your goal is to make it look like the other side