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  4. sin6A+cos6A+3sin2Acos2A is equal to

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Q. $ si{{n}^{6}}A+co{{s}^{6}}A+3si{{n}^{2}}Aco{{s}^{2}}A $ is equal to

Rajasthan PETRajasthan PET 2002

Solution:

$ si{{n}^{6}}A+co{{s}^{6}}A+3\text{ }si{{n}^{2}}A\text{ }co{{s}^{2}}A $
$ ={{({{\sin }^{2}}A)}^{3}}+{{({{\cos }^{2}}A)}^{3}}+3{{\sin }^{2}}A.{{\cos }^{2}}A $
$ =({{\sin }^{2}}A+{{\cos }^{2}}A)({{\sin }^{4}}A.+{{\cos }^{4}}A $
$ -{{\sin }^{2}}A{{\cos }^{2}}A)+3{{\sin }^{2}}A{{\cos }^{2}}A $
$ ={{({{\sin }^{2}}A)}^{2}}+{{({{\cos }^{2}}A)}^{2}}+2{{\sin }^{2}}A{{\cos }^{2}}A $
$ ={{({{\sin }^{2}}A+{{\cos }^{2}}A)}^{2}}=1 $