Question

how do prove this identity: sin^4x -cos^4x=1-2cos^2x

Answer 1

Do you still need help?

Answer 2

RS: (sin^2x+cos^2x)-2cos^2x
sin^2x-cos^2x
Thats how much i can simplify right side..

Answer 3

yah

Answer 4

alright.

Answer 5

So here is how I proved it:

Answer 6

So, change sinx^4 to (1-cosx^2)^2

Answer 7

You can expand that and you get 1-2cosx^2+cosx^4-cosx^4=1-2cosx^2

Answer 8

Do you follow so far?

Answer 9

@sabika13 Should I continue?

Answer 10

yes sorry

Answer 11

ok

Answer 12

When you simplify 1-2cosx^2+cosx^4-cosx^4=1-2cosx^2, the left side becomes 1-2cosx^2=1-2cosx^2

Answer 13

That proves the identity.

Answer 14

Normally, when you prove an identity, you work on one side. In this problem I worked on the left side.

Answer 15

wait how does (1-2cosx^2)^2=1-2cos^2+cosx^4?

Answer 16

To summarize it:

You have sin(x)^4-cos(x)^4=1-2cos(x)^2

change sin(x)^4 --->(1-cos(x)^2)^2

You have:
(1-cos(x)^2)^2-cos(x)^4=1-2cos(x)^2

Then expand (1-cos(x)^2)^2 on the left side.

You get

1-2cos(x)^2+cos(x)^4-cos(x)^4=1-2cos(x)^2

Answer 17

Any better?

Answer 18

Then the: cos(x)^4-cos(x)^4 cancel out.

Answer 19

Hope that helped.

Answer 20

im having problems expanding
(1-cos^2x)^2
shouldnt it be: 1-cos^4x :S

Answer 21

So, (1-cos^2x)(1-cos^2x). Just foil out.

Answer 22

ohhh okay, but why is it not: (1-cos^2x)(1+cos^2x)

Answer 23

It's squared. It's not a difference of squares.

Answer 24

ohh i think i get it...

Answer 25

Any number, or equation squared is the same as it times by itself.
Example (6)^2= (6)*(6) or (3x-2)^2= (3x-2)(3x-2). Do you get the point?

Answer 26

You don't change the signs inside the equation.

Answer 27

yeah so whats difference of squares then (if you dont mind me asking)

Answer 28

example x^2-4. When you factor it, you get (x-2)(x+2). It's a^2-b^2

Answer 29

ohhh thankkyou!!you helped me fix a very big misconception inmy head ;P

Answer 30

yep, no prob.