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Question Number 116006 by DELETED last updated on 30/Sep/20

$$3{(\sin x-\cos x)}^4+6{(\sin x+\cos x)}^2$$$$+4({\sin }^{6}x+{\cos }^{6}x)=\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}.$$

Answered by mindispower last updated on 30/Sep/20

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

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