Problem 20 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If in the triangle \(A B C, \tan \frac{A}{2}, \tan \frac{B}{2}\) and \(\tan \frac{C}{2}\) are in harmonic progression, then the least value of \(\cot \frac{B}{2}\) is (A) \(\sqrt{2}\) (B) \(\sqrt{3}\) (C) 2 (D) none of these

Step-by-Step Solution

Answer

The least value of \(\cot \frac{B}{2}\) is \(\sqrt{3}\).

Step 1: Understanding the Relationship in Harmonic Progression

Given that \(\tan \frac{A}{2}, \tan \frac{B}{2}, \tan \frac{C}{2}\) are in harmonic progression, recall that the reciprocals of these numbers are in arithmetic progression. Thus, \(\frac{1}{\tan \frac{A}{2}}, \frac{1}{\tan \frac{B}{2}}, \frac{1}{\tan \frac{C}{2}}\) form an arithmetic progression.

Step 2: Write the Arithmetic Progression Equation

For terms \(\frac{1}{\tan \frac{A}{2}}, \frac{1}{\tan \frac{B}{2}}, \frac{1}{\tan \frac{C}{2}}\) to be in an arithmetic progression, the middle term is the average of the terms on either side. Thus, \(2\cdot\frac{1}{\tan \frac{B}{2}} = \frac{1}{\tan \frac{A}{2}} + \frac{1}{\tan \frac{C}{2}}\).

Step 3: Understand the Relationship Between Angles in a Triangle

In triangle \(ABC\), the sum of angles is \(\pi\). So, \(A + B + C = \pi\). Correspondingly, for the half-angle identities, we have \(\tan \frac{A+B}{2} = \frac{1 - \tan \frac{A}{2} \tan \frac{B}{2}}{\tan \frac{A}{2} + \tan \frac{B}{2}} \).

Step 4: Use the Cotangent Identity

Utilize the fact that \(\cot \frac{B}{2} = \frac{1}{\tan \frac{B}{2}}\). Our goal now becomes finding the least value for \(\frac{1}{\tan \frac{B}{2}}\) given the arithmetic progression condition.

Step 5: Determine Minimum Cotangent Value

Solve the inequality: \(\tan \frac{B}{2}\) must be between \(\tan \frac{A}{2}\) and \(\tan \frac{C}{2}\). Assume symmetric values around 1 can minimize the expression using constraints from the triangle, leading to \(\cot \frac{B}{2} = \sqrt{3}\). B is optimally equilateral making all halves \(\frac{\pi}{6}\) leading \(\cot(\frac{\pi}{6}) = \sqrt{3}\).

Key Concepts

Harmonic Progression

Understanding harmonic progression is crucial when dealing with certain patterns in mathematics, especially in trigonometry related to triangles. A harmonic progression is a sequence of numbers where the reciprocals of the numbers form an arithmetic progression.
For example, if you have three numbers, say, \(a, b,\) and \(c\), in harmonic progression, their reciprocals \(\frac{1}{a}, \frac{1}{b},\) and \(\frac{1}{c}\) will form an arithmetic progression.

• The key characteristic is the relationship between the reciprocals.
• This concept is used to simplify complex trigonometric expressions, as it allows for easier calculation and understanding of connected values.

In the context of a triangle with angles \(\frac{A}{2}, \frac{B}{2},\) and \(\frac{C}{2}\), if the tangents of these half-angles are in a harmonic progression, it means their reciprocals follow a straightforward arithmetic pattern. This is an effective tool in finding relationships between the angles.

Triangles

Triangles are one of the fundamental shapes in geometry, consisting of three edges and three vertices. Every triangle has specific properties and characteristics that can aid in solving trigonometric problems.
In a triangle, the sum of the internal angles always equals \(\pi\) radians or 180 degrees. Let's recap some basic but crucial information about triangles:

• The three angles of any triangle, like \(A, B,\) and \(C\), should satisfy \(A + B + C = \pi\).
• Triangles can be classified based on their sides as equilateral, isosceles, or scalene, and based on their angles as acute, obtuse, or right.

The properties of triangles help in understanding half-angle identities and other trigonometric equations. Recognizing how the angles relate to each other is vital, particularly when they form a geometric or arithmetic relationship, like in harmonic progression.
This understanding helps in solving problems where you are asked to find specific trigonometric function values, such as the expression for \(\cot \frac{B}{2}\) in the given problem.

Half-Angle Identities

Half-angle identities are a set of trigonometric formulas that express the trigonometric functions of half angles in terms of square roots involving the full angles. They are very useful for solving equations involving angles that are not easily reducible by normal trigonometric identities.
These identities include:

• \( \tan \frac{A}{2} = \frac{1 - \cos A}{\sin A} = \frac{\sin A}{1 + \cos A} \)
• \( \cot \frac{A}{2} = \frac{1 + \cos A}{\sin A} = \frac{\sin A}{1 - \cos A} \)

In the case of the exercise, the half-angle identities for \(A, B,\) and \(C\) are applied to simplify and find relationships between the trigonometric functions of the triangle's angles. Such simplifications are essential in equations where you need the least or greatest value, as they may allow expressions to be reduced to more elementary forms.
The identity \(\tan \frac{A}{2}\) specifically comes in handy in determining the behavior of reciprocals of trigonometric functions, hence leading to solutions involving harmonic progressions of angles. This helps find the minimum value of functions like \(\cot \frac{B}{2}\) by exploiting symmetry in the triangle and the properties of the progression itself.