Question

sin^8 75° - cos^8 75°

Answer 1

$sin^875к-cos^875к=(sin^475к)^2-(cos^475к)^2=$$=(sin^475к-cos^475к)(sin^475к+cos^475к)=$$=(sin^275к-cos^275к)(sin^275к+cos^275к)(sin^475к+cos^475к)=$$=-(cos^275к-sin^275к)(sin^475к+cos^475к)=$$=-cos150к((sin^275к+cos^275к)^2-2sin^275cos^275к)=$$=cos30к(1^2- frac{1}{2} sin^2150)= frac{ sqrt{3} }{2} (1- frac{1}{2}*( frac{1}{2})^2)= frac{ sqrt{3} }{2} (1- frac{1}{2}* frac{1}{4})=$$=frac{ sqrt{3} }{2} (1- frac{1}{8})= frac{ sqrt{3} }{2} * frac{7}{8}= frac{7 sqrt{3} }{16}$