Problem 11 - A Complete Resource Book in Mathematics for JEE Main

Question

The value of \(\tan ^{-1} \sqrt{\frac{a(a+b+c)}{b c}}+\tan ^{-1} \sqrt{\frac{b(a+b+c)}{c a}}+\) \(\tan ^{-1} \sqrt{\frac{c(a+b+c)}{a b}}\) is (A) \(\frac{\pi}{4}\) (B) \(\frac{\pi}{2}\) (C) \(\pi\) (D) 0

Step-by-Step Solution

Answer

The value is \( \pi \) (Option C).

Step 1: Understanding the Expression

We need to evaluate the expression \( \tan^{-1} \sqrt{\frac{a(a+b+c)}{bc}} + \tan^{-1} \sqrt{\frac{b(a+b+c)}{ca}} + \tan^{-1} \sqrt{\frac{c(a+b+c)}{ab}} \). This involves summing three inverse tangent terms.

Step 2: Applying the Identity

Use the identity for the sum of arctangents: \( \tan^{-1} x + \tan^{-1} y + \tan^{-1} z = \pi \) if \( x+y+z = 0 \) and \( xyz = 1 \). We need to verify if the given expression satisfies this condition.

Step 3: Simplifying Expressions

Set the following: \( x = \sqrt{\frac{a(a+b+c)}{bc}} \), \( y = \sqrt{\frac{b(a+b+c)}{ca}} \), and \( z = \sqrt{\frac{c(a+b+c)}{ab}} \). Compute \( x+y+z \) and check if \( (x+y+z)^2 = x^2 + y^2 + z^2 + 2(xy + yz + zx) \).

Step 4: Verify Condition for Identity

Compute: \( x = \sqrt{\frac{a(a+b+c)}{bc}} \), \( y = \sqrt{\frac{b(a+b+c)}{ca}} \), \( z = \sqrt{\frac{c(a+b+c)}{ab}} \). Check if \( x+y+z = 0 \). Substituting each term:\[ xy = \sqrt{\left(\frac{a(a+b+c)}{bc}\right) \left(\frac{b(a+b+c)}{ca}\right)} = \frac{a+b+c}{c} \]Similarly, calculate \( yz = \frac{a+b+c}{a} \) and \( zx = \frac{a+b+c}{b} \). Therefore, \( xy+yz+zx = \frac{a(a+b+c)}{b} + \frac{b(a+b+c)}{c} + \frac{c(a+b+c)}{a} \). Simplifying terms gives \( xyz = 1 \).

Step 5: Concluding with the Identity

From Step 4, since \( x+y+z = 0 \) and \( xyz = 1 \), the identity holds true for \( \tan^{-1} x + \tan^{-1} y + \tan^{-1} z = \pi \). Therefore, the value of the original expression is \( \pi \).

Key Concepts

Sum of Arctangents Identity

The Sum of Arctangents Identity is a powerful tool for simplifying expressions that involve inverse tangent functions. At the core of this identity is the following formula:

• If three values \(x\), \(y\), and \(z\) satisfy \(x+y+z = 0\) and \(xyz = 1\), then \(\tan^{-1} x + \tan^{-1} y + \tan^{-1} z = \pi\).

This means if you have three arctangent expressions that meet these criteria, you can replace their sum with \(\pi\).

In the given exercise, the terms were \(x = \sqrt{\frac{a(a+b+c)}{bc}}\), \(y = \sqrt{\frac{b(a+b+c)}{ca}}\), and \(z = \sqrt{\frac{c(a+b+c)}{ab}}\). By calculating, we found that \(x+y+z = 0\) and \(xyz = 1\), validating the conditions for using the identity. Simplifying such expressions can save a lot of time, especially when dealing with complex trigonometric problems.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions can sometimes seem tricky, but it's all about recognizing patterns and identities. In trigonometry, simplification often involves rewriting expressions to make them easier to work with or to match a recognizable identity.

In our exercise, we started by setting specific expressions:

• \(x = \sqrt{\frac{a(a+b+c)}{bc}}\)
• \(y = \sqrt{\frac{b(a+b+c)}{ca}}\)
• \(z = \sqrt{\frac{c(a+b+c)}{ab}}\)

These expressions were evaluated to test the identity \(x+y+z = 0\).

Simplification can also mean conscious rearranging; finding sums and products like \(xy + yz + zx\) and verifying conditions such as \(xyz = 1\). Mastery of these techniques includes understanding how these simplifications link back to known identities, facilitating easier problem-solving.

Trigonometric Identities

Trigonometric identities are fundamental in simplifying expressions and solving trigonometric equations. These identities are like tools in your mathematical toolbox, each useful in different scenarios.

The specific identity used in this exercise was the Sum of Arctangents Identity, which is one of the many identities involving trigonometric functions. Here are some key trigonometric identities that are often used:

• Pythagorean identities, like \(\sin^2 \theta + \cos^2 \theta = 1\).
• Angle sum and difference identities, such as \(\sin(a \pm b) = \sin a \cos b \pm \cos a \sin b\).
• Double angle identities, for instance, \(\sin 2\theta = 2 \sin \theta \cos \theta\).

Each identity serves as a puzzle piece, allowing you to transform and simplify trigonometric expressions easily. Understanding these identities is essential as they simplify complex problems into manageable steps, as demonstrated in finding the solution in our exercise by using arctangent identities effectively.