We need to evaluate: \[ \cot^{-1}(21) + \cot^{-1}(13) + \cot^{-1}(-8) \] Step 1: Use the Identity
We use the identity: \[ \cot^{-1} A + \cot^{-1} B = \cot^{-1} \left(\frac{A B - 1}{A + B}\right) \] Applying this to \(\cot^{-1}(21)\) and \(\cot^{-1}(13)\): \[ \cot^{-1}(21) + \cot^{-1}(13) = \cot^{-1} \left(\frac{21 \times 13 - 1}{21 + 13} \right) \] \[ = \cot^{-1} \left(\frac{273 - 1}{34} \right) = \cot^{-1} \left(\frac{272}{34} \right) \] \[ = \cot^{-1}(8) \] Step 2: Use the Complementary Angle Identity
We use the identity: \[ \cot^{-1}(-x) = \pi - \cot^{-1}(x) \] So, \[ \cot^{-1}(-8) = \pi - \cot^{-1}(8) \] Step 3: Final Addition \[ \cot^{-1}(8) + \cot^{-1}(-8) = \cot^{-1}(8) + (\pi - \cot^{-1}(8)) = \pi \] Final Answer: \( \pi \)