Question

1:(sin^6a+cos^6a)





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Answer 1

$1:(sin^6a+cos^6a)=\frac{1}{sin^6a+cos^6a}=\frac{4}{4-3\, sin^22a}\\\\\\\sin^6a+cos^6a=(sin^2a)^3+(cos^2a)^3=\\\\=(\underbrace {sin^2a+cos^2a}_{1})(sin^4a-sin^2a\cdot cos^2a+cos^4a)=\\\\=(sin^4a+2sin^2a\cdot cos^2a+cos^4a)-2sin^2a\cdot cos^2a-sin^2a\cdot cos^2a=\\\\=(\underbrace {sin^2a+cos^2a}_{1})^2-3sin^2a\cdot cos^2a=1-3sin^2a\cdot cos^2a=\\\\=1-3\cdot (sina\cdot cosa)^2=1-3\cdot (\frac{1}{2}\, sin2a)^2=1-\frac{3}{4}\, sin^22a=\frac{4-3\, sin^22a}{4}$