Question

Вычислите
$sin^{4} frac{ pi }{12} + cos^{4} frac{ pi }{12}$

Answer 1

1-сosπ/6)²/4+(1+cosπ/6)²/4=(1-2cosπ/6+cos²π/6+1+2cosπ/6+cos²π/6)/4=
=(2+2cos²π/6)/4=(1+cos²π/6)/2=(1+3/4)/2=7/8

Answer 2

Удивительно, но Mathcard выдаёт ответ 5/8

Answer 3

$sin^4frac{pi}{12}+cos^4frac{pi}{12}=(sin^2frac{pi}{12})^2+(cos^2frac{pi}{12})^2=$

$=(frac{1-cosfrac{2pi}{12}}{2})^2+(frac{1+cosfrac{2pi}{12}}{2})^2=frac{(1-cosfrac{pi}{6})^2}{4}+frac{(1+cosfrac{pi}{6})^2}{4}=frac{(1-frac{sqrt3}{2})^2+(1+frac{sqrt3}{2})^2}{4}=\\=frac{1-sqrt3+frac{3}{4}+(1+sqrt3+frac{3}{4})}{4}=frac{1-sqrt3+frac{3}{4}+1+sqrt3+frac{3}{4}}{4}=frac{2+frac{6}{4}}{4}=frac{frac{14}{4}}{4}=frac{14}{16}=frac{7}{8}.$