Решение:

$$\frac{1}{\sqrt{37}-6} - \frac{1}{\sqrt{37}+6} = \frac{(\sqrt{37}+6) - (\sqrt{37}-6)}{(\sqrt{37}-6)(\sqrt{37}+6)} = \frac{\sqrt{37}+6 - \sqrt{37}+6}{(\sqrt{37})^2 - 6^2} = \frac{12}{37 - 36} = \frac{12}{1} = 12$$

Ответ: 12