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Prove the following identities: sin^8the...

Prove the following identities: `sin^8theta+cos^8theta=(sin^2theta-cos^2theta)(1-2sin^2thetacos^2theta)`

Text Solution

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L.H.S
`=sin^(8)theta - cos^(8)theta=(sin^(4)theta)^(2)-(cos^(4)theta)^(2)`
`=(sin^(4)theta-cos^(4)costheta)(sin^(4)theta+cos^(4)theta)`
`=[(sin^(2)theta)^(2)-(cos^(2)theta)^(2)][(sin^(2)theta)^(2)+(cos^(2)theta)^(2)]`
`=1.(sin^(2)theta-cos^(2)theta)(1^(2)-2sin^(2)thetacos^(2)theta)`
`=(sin^(2)theta-cos^(2)theta)(1-2sin^(2)thetacos^(2)theta)`
= RHS. Hence proved.
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