第13章 13.8

(1)

$\sin\frac{\pi}{9}+\sin\frac{2}{9}\pi=2\sin\frac{\dfrac{\pi}{9}+\dfrac{2}{9}\pi}{2}\cos\frac{\left(\dfrac{\pi}{9}-\dfrac{2}{9}\pi\right)}{2}=2\sin\frac{\pi}{6}\cos\frac{\pi}{18}$

(2)

$\cos\frac{2}{9}\pi+\cos\frac{7}{9}\pi=2\cos\frac{\dfrac{2}{9}\pi+\dfrac{7}{9}\pi}{2}\cos\frac{\dfrac{2}{9}\pi-\dfrac{7}{9}\pi}{2}=2\cos\frac{\pi}{2}\cos\frac{5}{18}\pi$

(3)

$\cos\frac{3}{4}\pi-\cos\frac{2}{3}\pi=-2\sin\frac{\dfrac{3}{4}\pi+\dfrac{2}{3}\pi}{2}\sin\frac{\dfrac{3}{4}\pi-\dfrac{2}{3}\pi}{2}=-2\sin\frac{17}{24}\pi\sin\frac{\pi}{24}$

(4)

$\sin8x-\sin4x=2\cos\frac{8x+4x}{2}\sin\frac{8x-4x}{2}=2\cos6x\cdot\sin2x$

(5)

$\cos2x+\cos4x=2\cos\frac{2x+4x}{2}\cos\frac{2x-4x}{2}=2\cos3x\cdot\cos x$

(6)

$\cos6x-\cos2x=-2\sin\frac{6x+2x}{2}\cdot\sin\frac{6x-2x}{2}=-2\sin4x\cdot\sin2x$