SOLUTION: verify the identity cos^4(x)+sin^4(x)=1-(1/2)sin(2x)^2

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Question 360885: verify the identity cos^4(x)+sin^4(x)=1-(1/2)sin(2x)^2
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
Work with the right side only:
`
Cos%5E4x%2BSin%5E4x+=+1-expr%281%2F2%29Sin%5E2%282x%29
`
Use the identity: red%28Sin%282theta%29=2Sin%28theta%29Cos%28theta%29%29
`
Cos%5E4x%2BSin%5E4x+=+1-expr%281%2F2%29%282Sin%28x%29Cos%28x%29%29%5E2
`
Square the expression in parentheses:
`
Cos%5E4x%2BSin%5E4x+=+1-expr%281%2F2%29%282%5E2Sin%5E2x%2ACos%5E2x%29
`
Cancel the 2 into the 2²:
`
Cos%5E4x%2BSin%5E4x+=+1-expr%281%2Fcross%282%29%29%282%5Ecross%282%29Sin%5E2x%2ACos%5E2x%29
`
Cos%5E4x%2BSin%5E4x+=+1-%282Sin%5E2x%2ACos%5E2x%29
`
This is a cool trick here. Replace 1 by 1²
`
Cos%5E4x%2BSin%5E4x+=+1%5E2-%282Sin%5E2x%2ACos%5E2x%29
`
Replace the 1 using the identity: red%28Sin%5E2theta+%2B+Cos%5E2theta+=+1%29
`
Cos%5E4x%2BSin%5E4x+=+%28Sin%5E2x%2BCos%5E2x%29%5E2-%282Sin%5E2x%2ACos%5E2x%29
`
Square out the first term on the right:
`

`
Remove the parentheses:
`
Cos%5E4x%2BSin%5E4x+=+Sin%5E4x%2B2Sin%5E2xCos%5E2x%2BCos%5E4x-2Sin%5E2x%2ACos%5E2x
`
Cancel the terms that cancel:
`

`
Cos%5E4x%2BSin%5E4x+=+Sin%5E4x%2BCos%5E4x
`
Edwin