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Prove the identity: `cos^4x -sin^4x=1+2sin^2 x` `` 

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nhl123 | Student, Grade 11 | Valedictorian

Posted November 15, 2011 at 7:24 AM via web

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Prove the identity:

`cos^4x -sin^4x=1+2sin^2 x`


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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted November 22, 2011 at 8:44 PM (Answer #1)

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Add `sin^4 x`  both sides.

`cos^4 x=1+2sin^2 x+sin^4 x`

Use the formula: `a^2+2ab+b^2=(a+b)^2`

Compare `1+2sin^2 x+sin^4 x`  and `a^2+2ab+b^2`  and notice that a=1 and b=sin^2 x

Restrict `1+2sin^2 x+sin^4 x`  using the formula.

`1+2sin^2 x+sin^4 x=(1+sin^2 x)^2`

Use the basic formula of trigonometry.

`sin^2 x+cos^2 x=1 =gtsin ^2 x = 1 - cos^2 x`

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

If you expand the square `(2-cos^2 x)^2`  you will obtain the result:

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

Notice that you did not get an identity.

ANSWER:  `cos^4 x - sin ^4 x!= 1+2sin^2 x.`

The expression  is not identity.

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