Question

Verify that (1-cos^2x)(1+cos^2x)=2sin^2x-sin^4x is a trig identity

Answer 1

By a certain Pythagorean identity we can say 1-cos^2(x)=?

Answer 2

Sin^2x?

Answer 3

yep! :)

Answer 4

so this means we now have
\[\sin^2(x)(1+\cos^2(x))=2\sin^2(x)-\sin^4(x)\]
now this is an identity we are trying to prove
now we have one side in terms of sin and another side in terms of mixture of sin and cos

Answer 5

We could use that same identity to rewrite cos^2(x)

Answer 6

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

Answer 7

that is a blank for you to fill in

Answer 8

I'm lost

Answer 9

Ok I want to rewrite cos^2(x) because the other side is just in terms of sin
I know an identity that has cos^2(x) and sin^2(x) in it.

Answer 10

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

Answer 11

So cos^2(x)=?

Answer 12

Sin^x+ 1

Answer 13

1-sin^2(x)

Answer 14

Sin^2x+1?

Answer 15

I was close

Answer 16

1-cos^2(x)=sin^2(x)
1-sin^2(x)=cos^2(x)
cos^2(x)+sin^2(x)=1

Answer 17

So we can back to what we are trying to prove and replace the cos^2(x) with 1-sin^2(x)
\[\sin^2(x)(1+(1-\sin^2(x))=2\sin^2(x)-\sin^4(x)\]

Answer 18

I replaced mr.cos^2(x) with mrs. (1-sin^2(x))

Answer 19

now we don't need those parenthesis since there is a + in front of that parenthesis
lets drop that extra stuff giving us:
\[\sin^2(x)(1+1-\sin^2(x))=2\sin^2(x)-\sin^4(x) \]

Answer 20

now 1+1=2 which I know you know (:p)
so now you distribute on that left hand side

Answer 21

I gtg I can figure out what's left. Friend me on Facebook @ Christopher McElhannon. I will need your help again :)

Answer 22

I will be here most likely. I'm not much of a facebook user. I'm on facebook free diet right now. :p