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Question
Prove the following identity :
`sin^8θ - cos^8θ = (sin^2θ - cos^2θ)(1 - 2sin^2θcos^2θ)`
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Solution
LHS = `sin^8θ - cos^8θ`
= `(sin^4θ)^2 - (cos^4θ)^2`
= `(sin^4θ - cos^4θ)(sin^4θ + cos^4θ)`
= `(sin^2θ - cos^2θ)(sin^2θ + cos^2θ)(sin^4θ + cos^4θ)`
= `(sin^2θ - cos^2θ)(sin^4θ + cos^4θ)`
= `(sin^2θ - cos^2θ)((sin^2θ)^2 + (cos^2θ)^2 + 2sin^2θcos^2θ - 2sin^2θcos^2θ)`
= `(sin^2θ - cos^2θ)((sin^2θ + cos^2θ)^2 - 2sin^2θcos^2θ)`
= `(sin^2θ - cos^2θ)(1 - 2sin^2θcos^2θ)`
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