Question

Упростите выражение

Answer 1

$1) 1-2\sin^2\alpha+\cos(4\alpha-2\pi)=\cos2\alpha+\cos(2\pi-4\alpha)=2\cos\frac{2\alpha+2\pi-4\alpha}{2}\cos \frac{2\alpha-2\pi+4\alpha}{2}=2\cos(\pi-\alpha)\cos(\pi-3\alpha)=2\cos\alpha\cos3\alpha.$

$2) \sin(\frac{\pi}{3}+\alpha) +\sin(\frac{\pi}{3}-\alpha)=2\sin\frac{\frac{\pi}{3}+\alpha+\frac{\pi}{3}-\alpha}{2}\cos \frac{\frac{\pi}{3}+\alpha-\frac{\pi}{3}+\alpha}{2}=2\sin\frac{\pi}{3}\cdot\cos\alpha=2\cdot\frac{\sqrt3}{2}\cdot\cos\alpha=\sqrt{3}\cos\alpha$