anonymous
  • anonymous
how do prove this identity: sin^4x -cos^4x=1-2cos^2x
Mathematics
jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
Do you still need help?
anonymous
  • anonymous
RS: (sin^2x+cos^2x)-2cos^2x sin^2x-cos^2x Thats how much i can simplify right side..
anonymous
  • anonymous
yah

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anonymous
  • anonymous
alright.
anonymous
  • anonymous
So here is how I proved it:
anonymous
  • anonymous
So, change sinx^4 to (1-cosx^2)^2
anonymous
  • anonymous
You can expand that and you get 1-2cosx^2+cosx^4-cosx^4=1-2cosx^2
anonymous
  • anonymous
Do you follow so far?
anonymous
  • anonymous
@sabika13 Should I continue?
anonymous
  • anonymous
yes sorry
anonymous
  • anonymous
ok
anonymous
  • anonymous
When you simplify 1-2cosx^2+cosx^4-cosx^4=1-2cosx^2, the left side becomes 1-2cosx^2=1-2cosx^2
anonymous
  • anonymous
That proves the identity.
anonymous
  • anonymous
Normally, when you prove an identity, you work on one side. In this problem I worked on the left side.
anonymous
  • anonymous
wait how does (1-2cosx^2)^2=1-2cos^2+cosx^4?
anonymous
  • anonymous
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
anonymous
  • anonymous
Any better?
anonymous
  • anonymous
Then the: cos(x)^4-cos(x)^4 cancel out.
anonymous
  • anonymous
Hope that helped.
anonymous
  • anonymous
im having problems expanding (1-cos^2x)^2 shouldnt it be: 1-cos^4x :S
anonymous
  • anonymous
So, (1-cos^2x)(1-cos^2x). Just foil out.
anonymous
  • anonymous
ohhh okay, but why is it not: (1-cos^2x)(1+cos^2x)
anonymous
  • anonymous
It's squared. It's not a difference of squares.
anonymous
  • anonymous
ohh i think i get it...
anonymous
  • anonymous
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?
anonymous
  • anonymous
You don't change the signs inside the equation.
anonymous
  • anonymous
yeah so whats difference of squares then (if you dont mind me asking)
anonymous
  • anonymous
example x^2-4. When you factor it, you get (x-2)(x+2). It's a^2-b^2
anonymous
  • anonymous
ohhh thankkyou!!you helped me fix a very big misconception inmy head ;P
anonymous
  • anonymous
yep, no prob.

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