Problem 87 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(\tan ^{-1} y=\tan ^{-1} x+\tan ^{-1}\left(\frac{2 x}{1-x^{2}}\right)\), where \(|x|<\frac{1}{\sqrt{3}}\), Then a value of \(y\) is : (A) \(\frac{3 x+x^{3}}{1-3 x^{2}}\) (B) \(\frac{3 x-x^{3}}{1+3 x^{2}}\) (C) \(\frac{3 x+x^{3}}{1+3 x^{2}}\) (D) \(\frac{3 x-x^{3}}{1-3 x^{2}}\)

Step-by-Step Solution

Answer

The value of \( y \) is \( \frac{3x - x^3}{1-3x^2} \), option (D).

Step 1: Understand the Arctan Addition Formula

The equation involves the addition of two arctan functions. Recall the identity: \[ \tan^{-1} a + \tan^{-1} b = \tan^{-1} \left( \frac{a + b}{1 - ab} \right) \] when \( ab < 1 \). This identity will be utilized to simplify the given expression.

Step 2: Assign Values to a and b

Let \( a = x \) and \( b = \frac{2x}{1-x^2} \). These are the values inside the arctan functions. Using the provided identity, we can express their sum as a single arctan term.

Step 3: Substitute and Simplify

Substitute \( a = x \) and \( b = \frac{2x}{1-x^2} \) into the formula: \[ \tan^{-1} y = \tan^{-1} \left( \frac{x + \frac{2x}{1-x^2}}{1 - x \cdot \frac{2x}{1-x^2}} \right) \]Simplify the expression by determining the numerator and denominator.

Step 4: Simplify the Numerator

Calculate the numerator:\[ x + \frac{2x}{1-x^2} \]Find a common denominator and simplify:\[ \frac{x(1-x^2) + 2x}{1-x^2} = \frac{x - x^3 + 2x}{1-x^2} = \frac{3x - x^3}{1-x^2} \]

Step 5: Simplify the Denominator

Calculate the denominator:\[ 1 - x \cdot \frac{2x}{1-x^2} \]Simplify this:\[ \frac{1-x^2 - 2x^2}{1-x^2} = \frac{1-3x^2}{1-x^2} \]

Step 6: Complete the Arctan Formula

Combine the simplified numerator and denominator: \[ \frac{3x - x^3}{1-x^2} \div \frac{1-3x^2}{1-x^2} = \frac{3x - x^3}{1-3x^2} \]Thus, \( \tan^{-1} y = \tan^{-1} \left( \frac{3x - x^3}{1-3x^2} \right) \) implies \( y = \frac{3x - x^3}{1-3x^2} \).

Step 7: Identify the Correct Option

With \( y = \frac{3x - x^3}{1-3x^2} \), compare with the given options. Therefore, the correct choice is option (D) \( \frac{3x - x^3}{1-3x^2} \).

Key Concepts

Arctan Addition Formula

The Arctan Addition Formula is a crucial tool when handling trigonometric expressions involving the arctan of sums. The formula states that for any two values, say \( a \) and \( b \), it holds that: \[ \tan^{-1} a + \tan^{-1} b = \tan^{-1} \left( \frac{a + b}{1 - ab} \right) \] This formula is valid when the product \( ab < 1 \), ensuring the arguments remain within the principal value range of the arctan function.
To see this in action, consider our exercise involving the variables \( x \) and \( \frac{2x}{1-x^2} \). When these expressions are substituted into the formula as values \( a \) and \( b \), it enables us to express their sum compactly under a single arctan function. This step is foundational in transforming the original problem into a simpler form that can be more easily worked with.

Trigonometric Identities

Trigonometric identities like the Arctan Addition Formula are powerful tools that simplify the process of working with trigonometric expressions. These identities represent established mathematical truths about trigonometric functions.
Identifying and applying the correct identity can greatly simplify the problem-solving process. Suppose you have to deal with the sum of two arctan expressions, as in the exercise at hand. Recognizing that the Arctan Addition Formula applies allows you to combine these into a singular expression:

• \( a = x \) and \( b = \frac{2x}{1-x^2} \) are substituted directly into the identity.
• This substitution transforms the sum into \( \tan^{-1} \left( \frac{x + \frac{2x}{1-x^2}}{1 - x \cdot \frac{2x}{1-x^2}} \right) \), simplifying the problem significantly.

Understanding these core trigonometric identities allows students to break down complex expressions into more manageable parts, making seemingly difficult problems much more approachable.

Problem Solving in Trigonometry

Engaging in problem solving with trigonometric functions requires strategic thinking and sometimes the use of clever substitutions or identities. The exercise we're discussing exemplifies this.
Initially, it poses a complex equation involving multiple arctan terms. The problem-solving path begins with:

• Breaking down the larger problem into subcomponents that involve simpler algebraic manipulations.
• Applying known trigonometric identities, like the Arctan Addition Formula, to converge the expression towards a simplified form.

Take the initial expression \( \tan^{-1} y = \tan^{-1} x + \tan^{-1} \left( \frac{2x}{1-x^2} \right) \). Through identification of applicable identities and systematic substitution, the problem is reduced to a more straightforward algebraic equation, making it easier to solve.

Simplification of Expressions

Simplifying expressions, especially those involving trigonometric functions, is an essential skill in mathematics. The ultimate goal is to make the expression as simple and understandable as possible.
Consider the expression from the task, which includes a complex fraction:

• For the numerator \( x + \frac{2x}{1-x^2} \), a common denominator is found to combine terms efficiently, resulting in \( \frac{3x - x^3}{1-x^2} \).
• For the denominator \( 1 - x \cdot \frac{2x}{1-x^2} \), simplifying yields \( \frac{1-3x^2}{1-x^2} \).

Combining these results, the expression becomes \[ \frac{3x - x^3}{1-3x^2} \]. Thus, through careful simplification, the task of solving the problem is considerably eased, guiding you directly to the correct answer choice.