Упр.23.4 ГДЗ Мерзляк 10 класс Базовый уровень (Алгебра)

1) $$\frac{\sin 80^\circ}{\cos 40^\circ}=\frac{\sin(2\cdot 40^\circ)}{\cos 40^\circ}=\frac{2\sin 40^\circ\cos 40^\circ}{\cos 40^\circ}=2\sin 40^\circ.$$

2) $$\cos 4\beta+\sin^2 2\beta=(\cos^2 2\beta-\sin^2 2\beta)+\sin^2 2\beta=\cos^2 2\beta.$$

3) $$\cos 6a+2\sin^2 3a=(\cos^2 3a-\sin^2 3a)+2\sin^2 3a=\cos^2 3a+\sin^2 3a=1.$$

4) $$\frac{\cos 70^\circ}{\cos 35^\circ+\sin 35^\circ}=\frac{\cos(2\cdot 35^\circ)}{\cos 35^\circ+\sin 35^\circ}=\frac{\cos^2 35^\circ-\sin^2 35^\circ}{\cos 35^\circ+\sin 35^\circ}$$

$$=\frac{(\cos 35^\circ-\sin 35^\circ)(\cos 35^\circ+\sin 35^\circ)}{\cos 35^\circ+\sin 35^\circ}=\cos 35^\circ-\sin 35^\circ.$$

5) $$\frac{1+\sin 2a}{(\sin a+\cos a)^2}=\frac{1+\sin 2a}{\sin^2 a+\cos^2 a+2\sin a\cos a}=\frac{1+\sin 2a}{1+\sin 2a}=1.$$

6) $$\sin a\cos a(\cos^2 a-\sin^2 a)=\frac12\sin 2a\cdot \cos 2a=\frac14\sin 4a.$$

7) $$\frac{\sin 4a}{\cos^4 a-\sin^4 a}=\frac{\sin 4a}{(\cos^2 a-\sin^2 a)(\cos^2 a+\sin^2 a)}=\frac{\sin 4a}{\cos 2a}$$

$$=\frac{2\sin 2a\cos 2a}{\cos 2a}=2\sin 2a.$$

8) $$\sin\left(\frac{\pi}{4}-a\right)\cos\left(\frac{\pi}{4}-a\right)=\frac12\sin\left(2\left(\frac{\pi}{4}-a\right)\right)=\frac12\sin\left(\frac{\pi}{2}-2a\right)=\frac12\cos 2a.$$

9) $$\sin^2(\beta-45^\circ)-\cos^2(\beta-45^\circ)=-\cos 2(\beta-45^\circ)=-\cos(2\beta-90^\circ)=-\sin 2\beta.$$

10) $$\frac{2\tg 1{,}5a}{1+\tg^2 1{,}5a}=\sin(2\cdot 1{,}5a)=\sin 3a.$$

Ответ: 1) $$2\sin 40^\circ$$; 2) $$\cos^2 2\beta$$; 3) $$1$$; 4) $$\cos 35^\circ-\sin 35^\circ$$; 5) $$1$$; 6) $$\frac14\sin 4a$$; 7) $$2\sin 2a$$; 8) $$\frac12\cos 2a$$; 9) $$-\sin 2\beta$$; 10) $$\sin 3a$$.