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