Question

Доказать, что 8cos⁴α-4cos³α-8cos²α+3cosα+1=-2sin(7α/2)*sin(α/2)

Answer 1

$8cos^4 alpha -4cos^3alpha -8cos^2alpha +3cosalpha +1=\ \ =-8cos^2alpha bigg(1-cos^2alpha bigg)-bigg(4cos^3alpha -3cosalpha bigg)+1=\ \ =-8cos^2alpha sin^2alpha -bigg(4cos^3alpha -3cosalpha bigg)+1=-2sin^22alpha -cos3alpha +1=\ \ =bigg(1-2sin^22alpha bigg)-cos3alpha =cos4alpha -cos3alpha =-2sin dfrac{7alpha }{2}sin dfrac{alpha }{2}$

Что и требовалось доказать.