Question

Упростить выражение
cos^2a-cos(a+ pi/6)cos(a-pi/6)

Answer 1

cos²α-cos(α+π/6)*cos(α-π/6)
cos²α-1/2(cos(α+π/6+a-π/6) + cos(α+π/6-α+π/6))
cos²α-1/2(cos2α+cos π/3) ;
cos²α-1/2cos2α-1/2*1/2 ;
cos²α - cos2α/2 - 1/4  ; домножим все на 4/4.
(4cos²α-2cos2α - 1)/4 = (4cos²α - 2(2cos²α-1) - 1 ) /4;
(4cos²α-4cos²α+2-1)/4 = 1/4

Answer 2

Спасибо

Answer 3

$cos^2 alpha -cos( alpha + frac{ pi }{6} )cos( alpha - frac{ pi }{6} )=cos^2 alpha - frac{1}{2}[cos( alpha + frac{ pi }{6} - alpha +frac{ pi }{6} )+$$+cos( alpha + frac{ pi }{6} + alpha -frac{ pi }{6} )]=cos^2 alpha - frac{1}{2}(cos frac{ pi }{3} +cos2 alpha )=$$=cos^2 alpha - frac{1}{2}cos frac{ pi }{3} - frac{1}{2} cos2 alpha =cos^2 alpha - frac{1}{2} * frac{1}{2}- frac{1}{2}cos2 alpha=$$=cos^2 alpha - frac{1}{4} - frac{1}{2}(2cos^2 alpha-1)= cos^2 alpha - frac{1}{4} -cos^2 alpha+ frac{1}{2}=- frac{1}{4} + frac{1}{2}= frac{1}{4}$

P. S.
$cosx*cosy= frac{1}{2} [cos(x-y)+cos(x+y)]$

$cos2 alpha =2cos^2 alpha -1$