Question Number 80219 by Power last updated on 01/Feb/20
Answered by som(math1967) last updated on 01/Feb/20
$${let}\frac{\pi}{\mathrm{7}}=\theta\: \\ $$$$\therefore{cos}\mathrm{2}\theta+{cos}\mathrm{4}\theta+{cos}\mathrm{6}\theta \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}{sin}\theta}\left(\mathrm{2}{cos}\mathrm{2}\theta{sin}\theta+\mathrm{2}{sin}\theta{cos}\mathrm{4}\theta\right. \\ $$$$\left.\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:+\mathrm{2}{sin}\theta{cos}\mathrm{6}\theta\right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}{sin}\theta}\left({sin}\mathrm{3}\theta−{sin}\theta+{sin}\mathrm{5}\theta−{sin}\mathrm{3}\theta\right. \\ $$$$\left.\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{sin}\mathrm{7}\theta−{sin}\mathrm{5}\theta\right) \\ $$$$\frac{\mathrm{1}}{\mathrm{2}{sin}\theta}\left({sin}\mathrm{7}\theta−{sin}\theta\right)\bigstar \\ $$$$=−\frac{\mathrm{1}}{\mathrm{2}{sin}\theta}×{sin}\theta=−\frac{\mathrm{1}}{\mathrm{2}}\:{ans} \\ $$$$\bigstar{sin}\mathrm{7}\theta={sin}\pi=\mathrm{0} \\ $$
Commented by peter frank last updated on 01/Feb/20
$${great} \\ $$
Commented by Power last updated on 02/Feb/20
$$\mathrm{thanks} \\ $$