The value of \( \cot^2 x + \csc^2 x \) when \( x = \frac{\pi}{4} \) is:

At \( x = \frac{\pi}{4} \), \( \cot(\frac{\pi}{4}) = 1 \) and \( \csc(\frac{\pi}{4}) = \sqrt{2} \). Thus, \( \cot^2(\frac{\pi}{4}) + \csc^2(\frac{\pi}{4}) = 1^2 + (\sqrt{2})^2 = 1 + 2 = 3 \) — wait, let's re-verify: \( 1 + 2 = 3 \). Option 3 is correct.