Упр.9.29 ГДЗ Никольский 10 класс (Алгебра)

а)

$$\sin\left(x+\frac{\pi}{4}\right)-\cos\left(x+\frac{\pi}{4}\right)$$
$$=\sin x\cos\frac{\pi}{4}+\cos x\sin\frac{\pi}{4}-\left(\cos x\cos\frac{\pi}{4}-\sin x\sin\frac{\pi}{4}\right)$$
$$=\frac{\sqrt2}{2}\sin x+\frac{\sqrt2}{2}\cos x-\frac{\sqrt2}{2}\cos x+\frac{\sqrt2}{2}\sin x =\sqrt2\,\sin x.$$

б)

$$2\cos\left(x-\frac{\pi}{3}\right)-2\sin\left(\frac{\pi}{3}+x\right)$$
$$=2\left(\cos x\cos\frac{\pi}{3}+\sin x\sin\frac{\pi}{3}\right)-2\left(\sin x\cos\frac{\pi}{3}+\cos x\sin\frac{\pi}{3}\right)$$
$$=2\left(\frac12\cos x+\frac{\sqrt3}{2}\sin x-\frac12\sin x-\frac{\sqrt3}{2}\cos x\right)$$
$$=\cos x-\sqrt3\cos x+\sqrt3\sin x-\sin x =(1-\sqrt3)(\cos x-\sin x).$$

Ответ

а) $$\sqrt2\,\sin x$$;
б) $$(1-\sqrt3)(\cos x-\sin x).$$