3(sin x-cos x)^{4}+6(sin x+cos x)^{2}+4(sin ^ | vedclass

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(B) Let $f(x) = 3(\sin x-\cos x)^{4}+6(\sin x+\cos x)^{2}+4(\sin ^{6} x+\cos ^{6} x)$.First,simplify each term:$3(\sin x-\cos x)^{4} = 3(1-2\sin x\cos

Vedclass · Accepted Answer

$13$

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(B) Let $f(x) = 3(\sin x-\cos x)^{4}+6(\sin x+\cos x)^{2}+4(\sin ^{6} x+\cos ^{6} x)$.First,simplify each term:$3(\sin x-\cos x)^{4} = 3(1-2\sin x\cos x)^{2} = 3(1+4\sin^{2}x\cos^{2}x-4\sin x\cos x) = 3+12\sin^{2}x\cos^{2}x-12\sin x\cos x$.$6(\sin x+\cos x)^{2} = 6(1+2\sin x\cos x) = 6+12\sin x\cos x$.$4(\sin^{6}x+\cos^{6}x) = 4(\sin^{2}x+\cos^{2}x)(\sin^{4}x+\cos^{4}x-\sin^{2}x\cos^{2}x) = 4(1)((\sin^{2}x+\cos^{2}x)^{2}-3\sin^{2}x\cos^{2}x) = 4(1-3\sin^{2}x\cos^{2}x) = 4-12\sin^{2}x\cos^{2}x$.Summing these up:$f(x) = (3+12\sin^{2}x\cos^{2}x-12\sin x\cos x) + (6+12\sin x\cos x) + (4-12\sin^{2}x\cos^{2}x)$.$f(x) = 3+6+4 + (12\sin^{2}x\cos^{2}x-12\sin^{2}x\cos^{2}x) + (-12\sin x\cos x+12\sin x\cos x) = 13$.

$3(\sin x-\cos x)^{4}+6(\sin x+\cos x)^{2}+4(\sin ^{6} x+\cos ^{6} x)$ is equal to

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