Problem 51 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(r=x+y+z\), then \(\tan ^{-1} \sqrt{\frac{x r}{y z}}+\tan ^{-1} \sqrt{\frac{y r}{z x}}+\tan ^{-1} \sqrt{\frac{z r}{x y}}=\) (A) \(\pi\) (B) \(2 \pi\) (C) \(\frac{\pi}{2}\) (D) none of these

Step-by-Step Solution

Answer

The answer is \(\pi\).

Step 1: Identify the Known Equation

The problem provides the expression for \(r\): \(r = x+y+z\). Recognize that \(r\) is the sum of the three variables \(x\), \(y\), and \(z\).

Step 2: Analyze the Trigonometric Expressions

We need to evaluate the sum \(\tan^{-1} \sqrt{\frac{x r}{yz}} + \tan^{-1} \sqrt{\frac{y r}{zx}} + \tan^{-1} \sqrt{\frac{z r}{xy}}\). Notice that each term inside the square root is a fraction, and each term involved corresponds to a variable multiplication with \(r\).

Step 3: Simplify the Inside of Trigonometric Functions

Substitute \(r = x + y + z\) into each of the fractions. For the first term, \(\frac{x(x+y+z)}{yz}\), simplify down to \(1 + \frac{x}{y} + \frac{x}{z}\), and follow similarly for the other two expressions.

Step 4: Use Trigonometric Addition Identities

Recognize that the sum \(\tan^{-1} A + \tan^{-1} B + \tan^{-1} C = \pi\) when \(A = \sqrt{\frac{xr}{yz}}\), \(B = \sqrt{\frac{yr}{zx}}\), and \(C = \sqrt{\frac{zr}{xy}}\). This is a known trigonometric identity when \(A, B, C\) satisfy certain conditional symmetry, which is true in this setup as they are cyclic in nature.

Step 5: Conclusion from Trigonometric Identity

Given the cyclic nature and structure of the problems, the identity applies, leading to the resulting sum \(\tan^{-1} \sqrt{\frac{x r}{yz}} + \tan^{-1} \sqrt{\frac{y r}{zx}} + \tan^{-1} \sqrt{\frac{z r}{xy}} = \pi\).

Key Concepts

Cyclic Symmetry

Cyclic symmetry involves repeating patterns where elements or operations have a cyclical sequence. In mathematics, particularly in trigonometry, equations or expressions sometimes have components that can be rearranged in a cyclic manner without changing the outcome. This concept is crucial when examining expressions that have similar structures and elements in different orders, just like the problem given.In the exercise, we're dealing with expressions involving three variables: \(x\), \(y\), and \(z\). Since these terms repeat in a specific cycle in each component of the trigonometric expression, they're said to have cyclic symmetry.

• The cyclic property lets us sum these components seamlessly, as they follow a predictable pattern.
• This helps simplify the problem-solving process and aids in recognizing trigonometric identities more easily.

By understanding that the variables interchangeably rotate positions yet maintain the relationship, you harness the power of cyclic symmetry, greatly simplifying your calculations.

Angle Addition Formulas

The angle addition formulas in trigonometry are essential tools for calculating the tangent, sine, or cosine of sum or difference of angles. For tangent, the formula is:\[\tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}\]In the exercise, we're tasked with finding the sum of three inverse tangent values. Recognizing the overall pattern and utilizing addition identities allow these complex expressions to boil down to more manageable forms.

• By breaking down each inverse tangent and applying the formula, you can transform and combine these angles into a single solution.
• This solution leverages the symmetry and cyclical nature of the problem, as the angles rotate and interact seamlessly without additional calculation difficulties.

The angle addition formulas become especially powerful when cyclic conditions are present, as the equation's very structure simplifies automatically when these identities apply.

Inverse Trigonometric Functions

Inverse trigonometric functions undo the work of regular trigonometric functions. If you know an angle's sine, cosine, or tangent, the inverse functions help find the angle itself. They are particularly useful when dealing with cyclically symmetric structures, such as the problem we're examining.For instance, \(\tan^{-1}(x)\) gives the angle whose tangent is \(x\). When tackling our problem:

• The expression involves multiple inverse tangent values, making it a perfect scenario to apply knowledge of inverse functions.
• When cyclic identity is discovered, it sets the stage for these inverse tangents to interact and add up to known constants, such as \(\pi\).

As these functions are often the inverse of real-world situations, recognizing their cyclic properties and being able to apply them correctly allows for simplification and the calculation of direct angle sums with ease.