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Question Number 118078 by bemath last updated on 15/Oct/20

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

Answered by bobhans last updated on 15/Oct/20

$$\left(1\right){(\sin \mathrm{x}+\cos \mathrm{x})}^4={(1+\sin 2\mathrm{x})}^2$$$$=1+2\sin 2\mathrm{x}+\sin {}^{2}2\mathrm{x}$$$$\left(2\right){(\sin \mathrm{x}-\cos \mathrm{x})}^2=1-\sin 2\mathrm{x}$$$$\left(3\right)\sin {}^6\mathrm{x}+\cos {}^6\mathrm{x}={\left(\sin {}^2\mathrm{x}\right)}^3+{\left(\cos {}^2\mathrm{x}\right)}^3$$$$=1(\sin {}^4\mathrm{x}-\sin {}^2\mathrm{xcos}{}^2\mathrm{x}+\cos {}^4\mathrm{x})$$$$={\left(\sin {}^2\mathrm{x}\right)}^2+{\left(\cos {}^2\mathrm{x}\right)}^2-\frac{1}{4}\sin {}^{2}2\mathrm{x}$$$$=1-2\sin {}^2\mathrm{xcos}{}^2\mathrm{x}-\frac{1}{4}\sin {}^{2}2\mathrm{x}$$$$=1-\frac{3}{4}\sin {}^{2}2\mathrm{x}$$$$\mathrm{Now}\mathrm{we}\mathrm{want}\mathrm{compute}3\times \left(1\right)+6\times \left(2\right)-3\times \left(3\right)$$$$\Rightarrow 3+6\sin 2\mathrm{x}+3\sin {}^{2}2\mathrm{x}+6-6\sin 2\mathrm{x}-$$$$(3-\frac{9}{4}\sin {}^{2}2\mathrm{x})=9+3\sin {}^{2}2\mathrm{x}-3+\frac{9}{4}\sin {}^{2}2\mathrm{x}$$$$=6+\frac{21}{4}\sin {}^{2}2\mathrm{x}$$

Answered by MJS_new last updated on 15/Oct/20

$$3{(c+s)}^4+6{(s-c)}^2-3(s^6+c^6)=$$$$=-3(c^6+s^6-c^4-s^4-4c^{3}s-4{cs}^3-6c^2{s}^2-2c^2-2s^2+4cs)=$$$$[c=\sqrt{1-s^2}]$$$$=-3(7(s^4-s^2)-2)=$$$$=3(7s^2(1-s^2)+2)=$$$$=3(7s^2{c}^2+2)=$$$$=6+{21\sin }^{2}x{\cos }^{2}x=$$$$=6+\frac{21}{4}{\sin }^{2}2x=$$$$=\frac{69}{8}-\frac{21}{8}\cos 4x$$

Commented by bemath last updated on 15/Oct/20

$$\mathrm{short}\mathrm{cut}\mathrm{way}$$

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