Multiple choice

If A, B, C, are angles of $\Delta ABC$ and $tan \frac{A}{2}, tan \frac{B}{2}, tan \frac{C}{2}$ are in H.P., then the minimum value of $cot \frac{B}{2}$ is

  1. $- \sqrt{3}$
  2. $ \sqrt{3}$
  3. $-2 \sqrt{3}$
  4. $2 \sqrt{3}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

If tan(A/2), tan(B/2), tan(C/2) are in HP, then 1/tan(A/2), 1/tan(B/2), 1/tan(C/2) are in AP, which means cot(A/2), cot(B/2), cot(C/2) are in AP. In any triangle, cot(A/2) + cot(C/2) = 2 * cot(B/2) * (something). Given the properties of triangles, the minimum value of cot(B/2) for this condition is sqrt(3).

AI explanation

Let the given tangents of half angles be a, b and c, where b represents tan of B by 2. Since they are in harmonic progression, the minimum value of cot of B by 2 occurs when the triangle is equilateral and each angle is 60 degrees. Therefore, B by 2 equals 30 degrees, and the cotangent of 30 degrees is the square root of 3.

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