Тригонометрия. Помогите пожалуйста, делаю ошибки, не получается. Докажите справедливость...

$tg\frac{\pi}{16}+tg\frac{3\pi}{16}+tg\frac{\pi}{16}*tg\frac{3\pi}{16}=\\\\ \frac{sin(\frac{\pi}{4})}{cos\frac{\pi}{16}*cos\frac{3\pi}{16}}+\frac{cos(\frac{\pi}{8})-cos(\frac{\pi}{4})}{cos(\frac{\pi}{8})+cos(\frac{\pi}{4})}=\\\\ \frac{\frac{\sqrt{2}}{2}}{\frac{cos\frac{\pi}{8}+cos\frac{\pi}{4}}{2}}+\frac{cos(\frac{\pi}{8})-cos(\frac{\pi}{4})}{cos(\frac{\pi}{8})+cos(\frac{\pi}{4})}=\\\\$
$\frac{\sqrt{2}}{cos\frac{\pi}{8}+cos\frac{\pi}{4}}+\frac{cos(\frac{\pi}{8})-cos(\frac{\pi}{4})}{cos(\frac{\pi}{8})+cos(\frac{\pi}{4})}=\\ \\ \frac{cos\frac{\pi}{8}+\frac{\sqrt{2}}{2}}{cos\frac{\pi}{8}+\frac{\sqrt{2}}{2}}=1$
верно

второе не верно вышло $tg\frac{7\pi}{7}-tg\frac{5\pi}{8}-tg\frac {7pi}{7}*tg\frac{5\pi}{8}=\\ -tg\frac{5\pi}{8}=ctg\frac{\pi}{8}$