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prove that sin^8x-cos^8x=(sin^2x-cos^2x)...

prove that `sin^8x-cos^8x=(sin^2x-cos^2x)(1-2sin^2xcos^2x)`

Text Solution

Verified by Experts

Using the property`a^2-b^2=(a-b)(a+b)`
=>`sin^(8)x-cos^(8)x=(sin^(2)x-cos^(2)x)(sin^(4)x+cos^(4)x)`.....(1)
we know=>`sin^(2)x+cos^(2)x=1`
squaring both sides we get,
`sin^(4)x+cos^(4)x=1-2sin^(2)xcos^(2)x`
substituting in one we get,
`sin^(8)x-cos^(8)x=(sin^(2)x-cos^(2)x)(1-2sin^(2)xcos^(2)x)`.....(1)
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