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

Trigonometric Functions

Solution:

$3(\sin x-\cos x)^{4}+6(\sin x+\cos x)^{2}+4\left(\sin ^{6} x+\cos ^{6} x\right)$
$=3(1-\sin 2 x)^{2}+6(1+\sin 2 x)+4\left[\left(\sin ^{2} x +\cos ^{2} x\right)^{3}-3 \sin ^{2} x \cos ^{2} x\left(\sin ^{2} x+\cos ^{2} x\right)\right]$
$=3\left(1-2 \sin 2 x+\sin ^{2} 2 x\right)+(6+6 \sin 2 x)+\left[1-\frac{3}{4} \sin ^{2} 2 x\right]$
$=13+3 \sin ^{2} 2 x-3 \sin ^{2} 2 x=13$