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To derive the reduction formula for I_n = ∫ cotⁿ x dx, we use the identity cot² x = csc² x - 1.I_n = ∫ cotⁿ x dx = ∫ cotⁿ⁻² x * cot² x dx= ∫ cotⁿ⁻² x (csc² x - 1) dx= ∫ cotⁿ⁻² x csc² x dx - ∫ cotⁿ⁻² x dxThe second term is simply I_n-2.For the first term, let u = cot x. Then du = -csc² x dx, so csc² x dx = -du.∫ cotⁿ⁻² x csc² x dx = ∫ uⁿ⁻² (-du) = -∫ uⁿ⁻² du= - (uⁿ⁻¹)/(n-1) + C = - (cotⁿ⁻¹ x)/(n-1) + C (for n ≠ 1).Combining these results, I_n = - (cotⁿ⁻¹ x)/(n-1) - I_n-2.