The value of \( i^{37} + \frac{1}{i^{67}} \) is:

Using \( i^4 = 1 \), \( i^{37} = i^{4\times 9 + 1} = i \) and \( \frac{1}{i^{67}} = i^{-67} = i^{-4\times 17 + 1} = i^{-3} = \frac{1}{-i} = i \). Thus, \( i + (-i) = 0 \).