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If ABC is a triangle and tan A/2, tan B/2, tan C/2 are in H.P., then the minimum value of cot B/2 is equal to
  • a)
    −√3
  • b)
    √3
  • c)
    Attains minimum if A = C
  • d)
    Attains minimum if 2A = C
Correct answer is option 'B,C'. Can you explain this answer?
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Most Upvoted Answer
If ABC is a triangle and tan A/2, tan B/2, tan C/2 are in H.P., then...
∵ In a triangle ABC, A + B + C = π
Hence equation (1)1 becomes
Equality holds when A = C
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If ABC is a triangle and tan A/2, tan B/2, tan C/2 are in H.P., then...
Given:
ABC is a triangle and tan(A/2), tan(B/2), tan(C/2) are in H.P.

To find:
The minimum value of cot(B/2)

Solution:
To find the minimum value of cot(B/2), we need to find the maximum value of tan(B/2). Since tan(B/2) is in H.P., we know that the sum of its conjugate is a constant.

Step 1: Express tan(B/2) in terms of other trigonometric ratios.
Using the formula for tan(A/2) = s/(s-a), where s is the semi-perimeter of the triangle and a is the length of side BC, we can express tan(B/2) and tan(C/2) in terms of other trigonometric ratios.

tan(B/2) = s/(s-c) ...(1)
tan(C/2) = s/(s-b) ...(2)

Step 2: Write the given condition in terms of tan(B/2) and tan(C/2).
Since tan(A/2), tan(B/2), tan(C/2) are in H.P., we can write the following equation:

2/tan(B/2) = 1/tan(A/2) + 1/tan(C/2)

Substituting the values of tan(A/2) and tan(C/2) from equations (1) and (2), we get:

2/tan(B/2) = (s-c)/s + (s-b)/s
2/tan(B/2) = (2s - b - c)/s
tan(B/2) = s/(2s - b - c) ...(3)

Step 3: Find the minimum value of cot(B/2).
The minimum value of cot(B/2) occurs when tan(B/2) is maximum. To find the maximum value of tan(B/2), we need to minimize the denominator in equation (3), which is (2s - b - c).

Since s is the semi-perimeter of the triangle and b, c are the lengths of sides AB and AC respectively, the minimum value of (2s - b - c) occurs when b = c.

Therefore, the minimum value of cot(B/2) is obtained when b = c, which implies that the triangle ABC is an isosceles triangle with base angles A = C.

Answer:
The minimum value of cot(B/2) is equal to √3.
The minimum value of cot(B/2) is obtained when A = C.
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If ABC is a triangle and tan A/2, tan B/2, tan C/2 are in H.P., then the minimum value of cot B/2 is equal toa)−√3b)√3c)Attains minimum if A = Cd)Attains minimum if 2A = CCorrect answer is option 'B,C'. Can you explain this answer?
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