584. Найдите значение выражения: 1) cos² 30° sin² 45°; 2) 3tg² 30° + 4tg 45° + cos 30° ctg 30°.

1) cos² 30° - sin² 45°

cos 30° = $$\frac{\sqrt{3}}{2}$$, sin 45° = $$\frac{\sqrt{2}}{2}$$

cos² 30° = $$\left( \frac{\sqrt{3}}{2} \right)^2$$ = $$\frac{3}{4}$$

sin² 45° = $$\left( \frac{\sqrt{2}}{2} \right)^2$$ = $$\frac{2}{4}$$ = $$\frac{1}{2}$$

cos² 30° - sin² 45° = $$\frac{3}{4} - \frac{1}{2} = \frac{3}{4} - \frac{2}{4} = \frac{1}{4}$$

Ответ: $$\frac{1}{4}$$

2) 3tg² 30° + 4tg 45° + cos 30° ctg 30°

tg 30° = $$\frac{\sqrt{3}}{3}$$, tg 45° = 1, cos 30° = $$\frac{\sqrt{3}}{2}$$, ctg 30° = $$\sqrt{3}$$

tg² 30° = $$\left( \frac{\sqrt{3}}{3} \right)^2$$ = $$\frac{3}{9}$$ = $$\frac{1}{3}$$

3tg² 30° = 3 × $$\frac{1}{3}$$ = 1

4tg 45° = 4 × 1 = 4

cos 30° ctg 30° = $$\frac{\sqrt{3}}{2} \cdot \sqrt{3} = \frac{3}{2}$$

3tg² 30° + 4tg 45° + cos 30° ctg 30° = 1 + 4 + $$\frac{3}{2}$$ = 5 + $$\frac{3}{2}$$ = $$\frac{10}{2} + \frac{3}{2} = \frac{13}{2}$$ = 6.5

Ответ: 6.5