Найдите в явном виде натуральное число, заданное выражением \(\left(\dfrac{2\mathop{\mathrm{tg}}\dfrac{\pi}{4}}{\sqrt{7}-\sqrt{3}}-\dfrac{2\mathop{\mathrm{ctg}}\dfrac{\pi}{4}}{\sqrt{7}+\sqrt{3}}\right)^2\).

Решение

Заметим, что \(\mathop{\mathrm{tg}}\dfrac{\pi}{4} = 1 = \mathop{\mathrm{ctg}}\dfrac{\pi}{4}.\)

Следовательно,

\[\left(\dfrac{2\mathop{\mathrm{tg}}\dfrac{\pi}{4}}{\sqrt{7}-\sqrt{3}}-\dfrac{2\mathop{\mathrm{ctg}}\dfrac{\pi}{4}}{\sqrt{7}+\sqrt{3}}\right)^2 = \left(\dfrac{2}{\sqrt{7}-\sqrt{3}}-\dfrac{2}{\sqrt{7}+\sqrt{3}}\right)^2 = \\ \]

\[= \left(\dfrac{2(\sqrt{7}+\sqrt{3})}{(\sqrt{7}-\sqrt{3})(\sqrt{7}+\sqrt{3})}-\dfrac{2(\sqrt{7}-\sqrt{3})}{(\sqrt{7}+\sqrt{3})(\sqrt{7}-\sqrt{3})}\right)^2 = \\ \]

\[= \left(\dfrac{2(\sqrt{7}+\sqrt{3})-2(\sqrt{7}-\sqrt{3})}{(\sqrt{7}-\sqrt{3})(\sqrt{7}+\sqrt{3})}\right)^2 = \left(\dfrac{2\sqrt{7}+2\sqrt{3}-2\sqrt{7}+2\sqrt{3}}{(\sqrt{7}-\sqrt{3})(\sqrt{7}+\sqrt{3})}\right)^2 = \\ \]

\[= \left(\dfrac{4\sqrt{3}}{(\sqrt{7})^2-(\sqrt{3})^2}\right)^2 = \left(\dfrac{4\sqrt{3}}{7-3}\right)^2 = \left(\dfrac{4\sqrt{3}}{4}\right)^2 = (\sqrt{3})^2 = 3.\]

Ответ: 3.