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Question Number 207516 by MATHEMATICSAM last updated on 17/May/24

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

Commented by A5T last updated on 17/May/24

$$Didyoueditthisquestiontochangesomething?$$

Answered by A5T last updated on 17/May/24

$$3{[{sin}^2\theta +{cos}^2\theta -2sin\theta cos\theta ]}^2+6(1+2sin\theta cos\theta )$$$$+4[{({sin}^2\theta +{cos}^2\theta )}^3-3{sin}^2\theta {cos}^2\theta ({sin}^2\theta +{cos}^2\theta )]$$$$=3{[1-2sin\theta cos\theta ]}^2+6+6sin2\theta +4[1-3{sin}^2\theta {cos}^2\theta ]$$$$=3[1+4{sin}^2\theta {cos}^2\theta -4sin\theta cos\theta ]+6+4$$$$-12{sin}^2\theta {cos}^2\theta +6sin2\theta$$$$=13-12sin\theta cos\theta +6sin2\theta =13$$

Commented by Frix last updated on 17/May/24

$$6{(\sin \theta +\cos \theta )}^2\ne 6{\left(1\right)}^2$$

Commented by A5T last updated on 17/May/24

$$6{({sin}^2\theta +{cos}^2\theta )}^{2}wasthereinitially,Isuppose.$$

Commented by Frix last updated on 17/May/24

$$\mathrm{I}{\mathrm{don}}^{\prime }\mathrm{t}\mathrm{know},\mathrm{I}\mathrm{saw}\mathrm{it}\mathrm{like}\mathrm{it}\mathrm{is}.$$$$\mathrm{But}\mathrm{I}\mathrm{believe}\mathrm{you},\mathrm{this}\mathrm{also}\mathrm{happened}\mathrm{to}\mathrm{me}$$$$\mathrm{before}...$$

Answered by Frix last updated on 17/May/24

$$\mathrm{Let}t=\tan \theta$$$$...=\frac{3{(t-1)}^4}{{(t^2+1)}^2}+\frac{6{(t+1)}^2}{t^2+1}+\frac{4(t^4-t^2+1)}{{(t^2+1)}^2}=13$$

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