Explanation

The expression 5sin θ + 12cos θ attains its maximum √(25 + 144) = 13 exactly when sin θ/5 = cos θ/12, i.e. sin θ = 5/13 and cos θ = 12/13. Then tan θ + cot θ = sin θ/cos θ + cos θ/sin θ = 1/(sin θcos θ) = 1/[(5/13)(12/13)] = 169/60, option B.