Упр.6.10 ГДЗ Мордкович 10-11 класс (Алгебра)

а) $$\operatorname{tg}\left(\frac{\pi}{5}\right)\cdot\operatorname{ctg}\left(\frac{\pi}{5}\right)$$; б) $$3\operatorname{tg}(2{,}3)\cdot\operatorname{ctg}(2{,}3)$$; в) $$\operatorname{tg}\left(\frac{\pi}{7}\right)\cdot\operatorname{ctg}\left(\frac{\pi}{7}\right)$$; г) $$7\operatorname{tg}\left(\frac{\pi}{12}\right)\cdot\operatorname{ctg}\left(\frac{\pi}{12}\right)$$.

Используем определения тангенса и котангенса:

$$\tg x=\frac{\sin x}{\cos x}, \qquad \ctg x=\frac{\cos x}{\sin x}.$$

1. $$\tg \frac{\pi}{5}\cdot \ctg \frac{\pi}{5}=\frac{\sin \frac{\pi}{5}}{\cos \frac{\pi}{5}}\cdot \frac{\cos \frac{\pi}{5}}{\sin \frac{\pi}{5}}=1.$$

2. $$3\tg 2{,}3\cdot \ctg 2{,}3=3\cdot \frac{\sin 2{,}3}{\cos 2{,}3}\cdot \frac{\cos 2{,}3}{\sin 2{,}3}=3.$$

3. $$\tg \frac{\pi}{7}\cdot \ctg \frac{\pi}{7}=\frac{\sin \frac{\pi}{7}}{\cos \frac{\pi}{7}}\cdot \frac{\cos \frac{\pi}{7}}{\sin \frac{\pi}{7}}=1.$$

4. $$7\tg \frac{\pi}{12}\cdot \ctg \frac{\pi}{12}=7\cdot \frac{\sin \frac{\pi}{12}}{\cos \frac{\pi}{12}}\cdot \frac{\cos \frac{\pi}{12}}{\sin \frac{\pi}{12}}=7.$$

Ответ: а) $$1$$; б) $$3$$; в) $$1$$; г) $$7$$.