475. Вычислить: 1) \(\cos\left(-\frac{\pi}{6}\right)\sin\left(-\frac{\pi}{3}\right)+\operatorname{tg}\left(-\frac{\pi}{4}\right)\); 2) \(\frac{1+\operatorname{tg}^2\left(-\frac{\pi}{6}\right)}{1+\operatorname{ctg}^2\left(-\frac{\pi}{6}\right)}\); 3) \(2\sin\left(-\frac{\pi}{6}\right)\cos\left(-\frac{\pi}{6}\right)+\operatorname{tg}\left(-\frac{\pi}{3}\right)+\sin^2\left(-\frac{\pi}{4}\right)\); 4) \(\cos(-\pi)+\operatorname{ctg}\left(-\frac{\pi}{2}\right)-\sin\left(-\frac{3\pi}{2}\right)+\operatorname{ctg}\left(-\frac{\pi}{4}\right)\); 5) \(\frac{3-\sin^2\left(-\frac{\pi}{3}\right)-\cos^2\left(-\frac{\pi}{3}\right)}{2\cos\left(-\frac{\pi}{4}\right)}\); 6) \(2\sin\left(-\frac{\pi}{6}\right)+3+7{,}5\operatorname{tg}(-\pi)+\frac{1}{8}\cos\frac{3\pi}{2}\).

1. Используем значения тригонометрических функций стандартных углов:

   \(\cos\left(-\frac{\pi}{6}\right)\sin\left(-\frac{\pi}{3}\right)+\operatorname{tg}\left(-\frac{\pi}{4}\right)=\frac{\sqrt{3}}{2}\cdot\left(-\frac{\sqrt{3}}{2}\right)-1=-\frac{3}{4}-1=-\frac{7}{4}.\)

2. \(\operatorname{tg}^2\left(-\frac{\pi}{6}\right)=\frac{1}{3}\), а \(\operatorname{ctg}^2\left(-\frac{\pi}{6}\right)=3\). Поэтому

   \(\frac{1+\operatorname{tg}^2\left(-\frac{\pi}{6}\right)}{1+\operatorname{ctg}^2\left(-\frac{\pi}{6}\right)}=\frac{1+\frac{1}{3}}{1+3}=\frac{4/3}{4}=\frac{1}{3}.\)

3. Подставляем значения функций:

   \(2\cdot\left(-\frac{1}{2}\right)\cdot\frac{\sqrt{3}}{2}-\sqrt{3}+\left(-\frac{\sqrt{2}}{2}\right)^2=-\frac{\sqrt{3}}{2}-\sqrt{3}+\frac{1}{2}=\frac{1-3\sqrt{3}}{2}.\)

4. \(\cos(-\pi)=-1\), \(\operatorname{ctg}\left(-\frac{\pi}{2}\right)=0\), \(\sin\left(-\frac{3\pi}{2}\right)=1\), \(\operatorname{ctg}\left(-\frac{\pi}{4}\right)=-1\). Тогда

   \(-1+0-1-1=-3.\)

5. По тождеству \(\sin^2 x+\cos^2 x=1\):

   \(\frac{3-\frac{3}{4}-\frac{1}{4}}{2\cdot\frac{\sqrt{2}}{2}}=\frac{2}{\sqrt{2}}=\sqrt{2}.\)

6. Так как \(\sin\left(-\frac{\pi}{6}\right)=-\frac{1}{2}\), \(\operatorname{tg}(-\pi)=0\), а \(\cos\frac{3\pi}{2}=0\), получаем

   \(2\cdot\left(-\frac{1}{2}\right)+3+7{,}5\cdot0+\frac{1}{8}\cdot0=-1+3=2.\)

Ответ: 1) \(-\frac{7}{4}\); 2) \(\frac{1}{3}\); 3) \(\frac{1-3\sqrt{3}}{2}\); 4) \(-3\); 5) \(\sqrt{2}\); 6) \(2\).