Упр.22.6 ГДЗ Мордкович 10-11 класс (Алгебра)

• а) $$\operatorname{tg}25+\operatorname{tg}35$$;
• б) $$\operatorname{tg}\frac{\pi}{5}-\operatorname{tg}\frac{\pi}{10}$$;
• в) $$\operatorname{tg}20+\operatorname{tg}40$$;
• г) $$\operatorname{tg}\frac{\pi}{3}-\operatorname{tg}\frac{\pi}{4}$$.

а) $$\tg 25^\circ+\tg 35^\circ=\frac{\sin 25^\circ}{\cos 25^\circ}+\frac{\sin 35^\circ}{\cos 35^\circ}=\frac{\sin 25^\circ\cos 35^\circ+\cos 25^\circ\sin 35^\circ}{\cos 25^\circ\cos 35^\circ}$$

$$=\frac{\sin(25^\circ+35^\circ)}{\cos 25^\circ\cos 35^\circ}=\frac{\sin 60^\circ}{\cos 25^\circ\cos 35^\circ}=\frac{\sqrt{3}}{2\cos 25^\circ\cos 35^\circ}.$$

б) $$\tg \frac{\pi}{5}-\tg \frac{\pi}{10}=\frac{\sin \frac{\pi}{5}}{\cos \frac{\pi}{5}}-\frac{\sin \frac{\pi}{10}}{\cos \frac{\pi}{10}}=\frac{\sin \frac{\pi}{5}\cos \frac{\pi}{10}-\cos \frac{\pi}{5}\sin \frac{\pi}{10}}{\cos \frac{\pi}{5}\cos \frac{\pi}{10}}$$

$$=\frac{\sin\left(\frac{\pi}{5}-\frac{\pi}{10}\right)}{\cos \frac{\pi}{5}\cos \frac{\pi}{10}}=\frac{\sin \frac{\pi}{10}}{\cos \frac{\pi}{5}\cos \frac{\pi}{10}}=\frac{\tg \frac{\pi}{10}}{\cos \frac{\pi}{5}}.$$

в) $$\tg 20^\circ+\tg 40^\circ=\frac{\sin 20^\circ}{\cos 20^\circ}+\frac{\sin 40^\circ}{\cos 40^\circ}=\frac{\sin 20^\circ\cos 40^\circ+\cos 20^\circ\sin 40^\circ}{\cos 20^\circ\cos 40^\circ}$$

$$=\frac{\sin(20^\circ+40^\circ)}{\cos 20^\circ\cos 40^\circ}=\frac{\sin 60^\circ}{\cos 20^\circ\cos 40^\circ}=\frac{\sqrt{3}}{2\cos 20^\circ\cos 40^\circ}.$$

г) $$\tg \frac{\pi}{3}-\tg \frac{\pi}{4}=\frac{\sin \frac{\pi}{3}}{\cos \frac{\pi}{3}}-\frac{\sin \frac{\pi}{4}}{\cos \frac{\pi}{4}}=\frac{\sin \frac{\pi}{3}\cos \frac{\pi}{4}-\cos \frac{\pi}{3}\sin \frac{\pi}{4}}{\cos \frac{\pi}{3}\cos \frac{\pi}{4}}$$

$$=\frac{\sin\left(\frac{\pi}{3}-\frac{\pi}{4}\right)}{\cos \frac{\pi}{3}\cos \frac{\pi}{4}}=\frac{\sin \frac{\pi}{12}}{\cos \frac{\pi}{3}\cos \frac{\pi}{4}}=\frac{\sin \frac{\pi}{12}}{\frac12\cdot \frac{\sqrt2}{2}}=2\sqrt2\,\sin \frac{\pi}{12}.$$