11. Найдите значение выражения 18√2 cos²(7)-√162.

11. Найдем значение выражения $$18\sqrt{2}\cos^2(\frac{7\pi}{8}) - \sqrt{162}$$.

$$18\sqrt{2}\cos^2(\frac{7\pi}{8}) - \sqrt{162} = 18\sqrt{2}\cos^2(\frac{7\pi}{8}) - \sqrt{81\cdot 2} = 18\sqrt{2}\cos^2(\frac{7\pi}{8}) - 9\sqrt{2}$$

$$18\sqrt{2}\cos^2(\frac{7\pi}{8}) - 9\sqrt{2} = 9\sqrt{2} (2\cos^2(\frac{7\pi}{8}) - 1) = 9\sqrt{2} \cos(2\cdot \frac{7\pi}{8}) = 9\sqrt{2}\cos(\frac{7\pi}{4})$$

$$9\sqrt{2}\cos(\frac{7\pi}{4}) = 9\sqrt{2}\cos(315^\circ) = 9\sqrt{2}\cos(360^\circ - 45^\circ) = 9\sqrt{2} \cos 45^\circ = 9\sqrt{2} \cdot \frac{\sqrt{2}}{2} = 9$$

Ответ: 9