Question

Упростите выражение:

Answer 1

$Sin(\pi+\frac{\alpha }{2})Cos(\frac{\pi }{2}-\frac{5\alpha }{2})-\frac{1}{2}(1-3Cos\alpha)-\frac{1}{4}(Cos3\alpha+3Cos\alpha)=-Sin\frac{\alpha }{2}Sin\frac{5\alpha }{2} -\frac{1}{2}+\frac{3}{2} Cos\alpha-\frac{1}{4}(4Cos^{3}\alpha-3Cos\alpha+3Cos\alpha)=-\frac{1}{2}[Cos(\frac{\alpha }{2}-\frac{5\alpha }{2})-Cos(\frac{\alpha }{2}+\frac{5\alpha }{2})]-\frac{1}{2}+\frac{3}{2}Cos\alpha-Cos^{3}\alpha =-\frac{1}{2}Cos2\alpha+\frac{1}{2}Cos3\alpha-\frac{1}{2}+\frac{3}{2}Cos\alpha$$-Cos^{3} \alpha=-\frac{1}{2}(2Cos^{2}\alpha-1)+\frac{1}{2}(4Cos^{3}\alpha-3Cos\alpha)-\frac{1}{2}+\frac{3}{2}Cos\alpha-Cos^{3}\alpha=-Cos^{2}\alpha+\frac{1}{2}+2Cos^{3}\alpha-\frac{3}{2}Cos\alpha-\frac{1}{2}+\frac{3}{2}Cos\alpha-Cos^{3}\alpha =Cos^{3}\alpha-Cos^{2}\alpha=Cos^{2}\alpha(Cos\alpha-1)\\\\Otvet:\boxed{Cos^{2}\alpha(Cos\alpha-1)}$