If a > 0, b > 0, a + B = 1, the square of (a + 1 / a) + (B + 1 / b) is greater than or equal to 25 / 2 Use sine and cosine functions to solve the problem

If a > 0, b > 0, a + B = 1, the square of (a + 1 / a) + (B + 1 / b) is greater than or equal to 25 / 2 Use sine and cosine functions to solve the problem

(a + 1 / a) ^ 2 + (B + 1 / b) ^ 2 = a ^ 2 + 1 / A ^ 2 + 2 + B ^ 2 + 1 / b ^ 2 + 2 = (a ^ 2 + B ^ 2) + (1 / A ^ 2 + 1 / b ^ 2) + 4 ≥ 0.5 * (a + b) ^ 2 + 0.5 * (1 / A + 1 / b) ^ 2 + 4 = 0.5 + 0.5 * (1 / A + 1 / b) ^ 2 + 4 = 4.5 + 0.5 * (1 / A + 1 / b) ^ 2 because ab ≤ 0.25 * (a + b) ^ 2 = 0.25, so 1 / A + 1 / b = (a + b)

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