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Prove that: 3 (sin x-cos x)^4+ 6 (sin x ...

Prove that: `3 (sin x-cos x)^4+ 6 (sin x +cosx)^ 2+4 (sin^6 x+ cos^6 x) -13=0`

Text Solution

Verified by Experts

The correct Answer is:
C
Given expression =
`3(sinx-cosx)^(4) + 6(sinx + cosx)^(2) + 4(sin^(6)x + cos^(6)x)`
`=3(1-sin2x)^(2) + 6(1 + sin2x) + 4{(sin^(2) x+cos^(2)x)^(3) - 3 sin^(2)x cos^(2)x(sin^(2)x+cos^(2)x)}`
`=3(1-2sin 2x + sin^(2)2x)+6+6sin2x+4(1-3sin^(2)x cos^(2)x)`
`=3(1-2 sin 2x + sin^(2)2x + 2 +2sin2x)+4(1-(3)/(4).sin^(2)2x)`
`=13+3sin^(2)2x - 3 sin^(2)2x =13`
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