Explanation

With t = tan²x > 0: 25csc²x + sec²x = 25(1 + 1/t) + (1 + t) = 26 + t + 25/t. By AM–GM, t + 25/t ≥ 2√25 = 10, so the minimum value is 36 (attained at tan²x = 5). Equality in AM–GM holds when t = 25/t, i.e. tan²x = 5, which is attainable.