Question

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$sinfrac{pi}{12}*(cos^{6}frac{pi}{24}-sin^{6}frac{pi}{24})=$

Answer 1

 

$sinfrac{pi}{12}*(cos^{6}frac{pi}{24}-sin^{6}frac{pi}{24}) =\\ sinfrac{pi}{12}*((cos^{3}frac{pi}{24})^2-(sin^{3}frac{pi}{24})^2) =\\ sinfrac{pi}{12}*(cos^{3}frac{pi}{24}-sin^{3}frac{pi}{24})(cos^{3}frac{pi}{24}+sin^{3}frac{pi}{24}) =$

 

$sinfrac{pi}{12}*(cosfrac{pi}{24}-sinfrac{pi}{24})(cos^2frac{pi}{24} + cosfrac{pi}{24}sinfrac{pi}{24} + sin^2frac{pi}{24})*\\ (cosfrac{pi}{24}+sinfrac{pi}{24})(cos^2frac{pi}{24} - cosfrac{pi}{24}sinfrac{pi}{24} + sin^2frac{pi}{24}) =\\ sinfrac{pi}{12}*(cosfrac{pi}{24}-sinfrac{pi}{24})(1 + cosfrac{pi}{24}sinfrac{pi}{24})*\\ (cosfrac{pi}{24}+sinfrac{pi}{24})(1 - cosfrac{pi}{24}sinfrac{pi}{24}) =$

 

$sinfrac{pi}{12}(cos^2frac{pi}{24}-sin^2frac{pi}{24})(1 + cosfrac{pi}{24}sinfrac{pi}{24}) (1 - cosfrac{pi}{24}sinfrac{pi}{24}) = \\ sinfrac{pi}{12}(cos^2frac{pi}{24}-sin^2frac{pi}{24})(1 - cos^2frac{pi}{24}sin^2 frac{pi}{24}) =$

 

$[cos2x = cos^2x - sin^2x]\\ sinfrac{pi}{12}cosfrac{pi}{12}(1 - cos^2frac{pi}{24}sin^2frac{pi}{24}) = \\ frac{1}{2}sinfrac{pi}{6}(1 - frac{1}{4}(sin^2frac{pi}{12})) </var>=$</p> <p> </p> <p> </p> <p><img src=[/tex]frac{1}{2}sinfrac{pi}{6}(1 - frac{1}{4}(frac{1 - cosfrac{pi}{6}}{2})) = frac{1}{2}*frac{1}{2}*(1 - frac{1}{8}(1 - frac{sqrt{3}}{2})) =\\ frac{1}{4}(1 - frac{1}{8} + frac{sqrt{3}}{16})= frac{1}{4}*(frac{14+sqrt{3}}{16}) = boxed{frac{14 + sqrt{3}}{64}}" title="[cos2x = cos^2x - sin^2x]\\ sinfrac{pi}{12}cosfrac{pi}{12}(1 - cos^2frac{pi}{24}sin^2frac{pi}{24}) = \\ frac{1}{2}sinfrac{pi}{6}(1 - frac{1}{4}(sin^2frac{pi}{12})) = " title="frac{1}{2}sinfrac{pi}{6}(1 - frac{1}{4}(frac{1 - cosfrac{pi}{6}}{2})) = frac{1}{2}*frac{1}{2}*(1 - frac{1}{8}(1 - frac{sqrt{3}}{2})) =\\ frac{1}{4}(1 - frac{1}{8} + frac{sqrt{3}}{16})= frac{1}{4}*(frac{14+sqrt{3}}{16}) = boxed{frac{14 + sqrt{3}}{64}}" title="[cos2x = cos^2x - sin^2x]\\ sinfrac{pi}{12}cosfrac{pi}{12}(1 - cos^2frac{pi}{24}sin^2frac{pi}{24}) = \\ frac{1}{2}sinfrac{pi}{6}(1 - frac{1}{4}(sin^2frac{pi}{12})) = " alt="frac{1}{2}sinfrac{pi}{6}(1 - frac{1}{4}(frac{1 - cosfrac{pi}{6}}{2})) = frac{1}{2}*frac{1}{2}*(1 - frac{1}{8}(1 - frac{sqrt{3}}{2})) =\\ frac{1}{4}(1 - frac{1}{8} + frac{sqrt{3}}{16})= frac{1}{4}*(frac{14+sqrt{3}}{16}) = boxed{frac{14 + sqrt{3}}{64}}" title="[cos2x = cos^2x - sin^2x]\\ sinfrac{pi}{12}cosfrac{pi}{12}(1 - cos^2frac{pi}{24}sin^2frac{pi}{24}) = \\ frac{1}{2}sinfrac{pi}{6}(1 - frac{1}{4}(sin^2frac{pi}{12})) = " />

 

 

$[cos2x = cos^2x - sin^2x]\\ sinfrac{pi}{12}cosfrac{pi}{12}(1 - cos^2frac{pi}{24}sin^2frac{pi}{24}) = \\ frac{1}{2}sinfrac{pi}{6}(1 - frac{1}{4}(sin^2frac{pi}{12})) </var>=$

 

 

[tex]frac{1}{2}sinfrac{pi}{6}(1 - frac{1}{4}(frac{1 - cosfrac{pi}{6}}{2})) = frac{1}{2}*frac{1}{2}*(1 - frac{1}{8}(1 - frac{sqrt{3}}{2})) =\\ frac{1}{4}(1 - frac{1}{8} + frac{sqrt{3}}{16})= frac{1}{4}*(frac{14+sqrt{3}}{16}) = boxed{frac{14 + sqrt{3}}{64}}" />