Упр.1094 ГДЗ Колягин Ткачёва 10 класс (Алгебра)

1) sin(пи/3+а) + sin(пи/3-a);
2) cos(пи/4-b)-cos(пи/4 + b)
3) sin^2(пи/4+а) + sin^2(пи/4-a);
4) cos^2(a-пи/4)-cos^2(a+пи/4)

1. $$\sin\left(\frac{\pi}{3}+a\right)+\sin\left(\frac{\pi}{3}-a\right)=2\sin\frac{\frac{\pi}{3}+a+\frac{\pi}{3}-a}{2}\cos\frac{\frac{\pi}{3}+a-\left(\frac{\pi}{3}-a\right)}{2}$$
   $$=2\sin\frac{\pi}{3}\cos a=2\cdot\frac{\sqrt{3}}{2}\cos a=\sqrt{3}\cos a.$$

2. $$\cos\left(\frac{\pi}{4}-b\right)-\cos\left(\frac{\pi}{4}+b\right)=-2\sin\frac{\frac{\pi}{4}-b+\frac{\pi}{4}+b}{2}\sin\frac{\frac{\pi}{4}-b-\left(\frac{\pi}{4}+b\right)}{2}$$
   $$=-2\sin\frac{\pi}{4}\sin(-b)=2\sin\frac{\pi}{4}\sin b=2\cdot\frac{\sqrt{2}}{2}\sin b=\sqrt{2}\sin b.$$

3. $$\sin^2\left(\frac{\pi}{4}+a\right)-\sin^2\left(\frac{\pi}{4}-a\right)$$
   $$=\left(\sin\left(\frac{\pi}{4}+a\right)-\sin\left(\frac{\pi}{4}-a\right)\right)\left(\sin\left(\frac{\pi}{4}+a\right)+\sin\left(\frac{\pi}{4}-a\right)\right)$$
   $$=2\cos\frac{\frac{\pi}{4}+a+\frac{\pi}{4}-a}{2}\sin\frac{\left(\frac{\pi}{4}+a\right)-\left(\frac{\pi}{4}-a\right)}{2}\cdot 2\sin\frac{\frac{\pi}{4}+a+\frac{\pi}{4}-a}{2}\cos\frac{\left(\frac{\pi}{4}+a\right)-\left(\frac{\pi}{4}-a\right)}{2}$$
   $$=2\cos\frac{\pi}{4}\sin a\cdot 2\sin\frac{\pi}{4}\cos a=4\cdot\frac{\sqrt{2}}{2}\cdot\frac{\sqrt{2}}{2}\sin a\cos a$$
   $$=2\sin a\cos a=\sin 2a.$$

4. $$\cos^2\left(a-\frac{\pi}{4}\right)-\cos^2\left(a+\frac{\pi}{4}\right)$$
   $$=\left(\cos\left(a-\frac{\pi}{4}\right)-\cos\left(a+\frac{\pi}{4}\right)\right)\left(\cos\left(a-\frac{\pi}{4}\right)+\cos\left(a+\frac{\pi}{4}\right)\right)$$
   $$=-2\sin\frac{\left(a-\frac{\pi}{4}\right)+\left(a+\frac{\pi}{4}\right)}{2}\sin\frac{\left(a-\frac{\pi}{4}\right)-\left(a+\frac{\pi}{4}\right)}{2}$$
   $$\cdot 2\cos\frac{\left(a-\frac{\pi}{4}\right)+\left(a+\frac{\pi}{4}\right)}{2}\cos\frac{\left(a-\frac{\pi}{4}\right)-\left(a+\frac{\pi}{4}\right)}{2}$$
   $$=-2\sin a\sin\left(-\frac{\pi}{4}\right)\cdot 2\cos a\cos\left(-\frac{\pi}{4}\right)$$
   $$=4\sin a\cos a\cdot\sin\frac{\pi}{4}\cos\frac{\pi}{4}=4\sin a\cos a\cdot\frac{\sqrt{2}}{2}\cdot\frac{\sqrt{2}}{2}$$
   $$=2\sin a\cos a=\sin 2a.$$

Ответ

1) $$\sqrt{3}\cos a$$; 2) $$\sqrt{2}\sin b$$; 3) $$\sin 2a$$; 4) $$\sin 2a$$.