Question

cos2альфа+2sin(альфа+пи/6)sin(альфа-пи/6)

Answer 1

$=cos2 alpha +2(sin alpha cos frac{ pi }{6}+sin frac{ pi }{6} cos alpha )(sin alpha cos frac{ pi }{6} -sin frac{ pi }{6}cos alpha )=$$cos2 alpha +2(frac{ sqrt{3} }{2} sin alpha + frac{1}{2} cos alpha )(frac{ sqrt{3} }{2} sin alpha -frac{1}{2} cos alpha )=$$cos2 alpha +2( frac{3}{4}sin^2 alpha - frac{1}{4}cos^2 alpha )=cos2 alpha + frac{1}{2}(3sin^2 alpha -cos^2 alpha )=$$cos2 alpha + frac{1}{2}(2sin^2 alpha -cos2 alpha )= cos2 alpha + frac{1}{2}(1-cos2 alpha -cos2 alpha )=$$cos2 alpha + frac{1}{2} -cos2 alpha = frac{1}{2}$