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

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 B).

Step 1: Understand the Relationship

Given the equation \( \tan^{-1} y = \tan^{-1} x + \tan^{-1}\left(\frac{2x}{1-x^2}\right) \), we want a single expression for \( y \) using the formula for the sum of inverse tangents: \[ \tan^{-1} a + \tan^{-1} b = \tan^{-1}\left( \frac{a+b}{1-ab} \right) \] when \( ab < 1 \).

Step 2: Apply the Addition Formula

Let \( a = x \) and \( b = \frac{2x}{1-x^2} \). Using the tangent addition formula, we compute: \[ \tan^{-1} y = \tan^{-1}\left( \frac{x + \frac{2x}{1-x^2}}{1 - x\cdot\frac{2x}{1-x^2}} \right) \].

Step 3: Simplify the Numerator

Simplify the numerator: \( x + \frac{2x}{1-x^2} \). Combining terms yields: \[ \frac{x(1-x^2) + 2x}{1-x^2} = \frac{x - x^3 + 2x}{1-x^2} = \frac{3x - x^3}{1-x^2} \].

Step 4: Simplify the Denominator

Simplify the denominator: \( 1 - x \cdot \frac{2x}{1-x^2} = \frac{1 - 2x^2}{1-x^2} \).

Step 5: Combine and Solve for y

Now, the entire expression for \( y \) using the identity becomes: \[ \tan^{-1} y = \tan^{-1}\left( \frac{\frac{3x - x^3}{1-x^2}}{\frac{1 - 2x^2}{1-x^2}} \right) = \tan^{-1}\left( \frac{3x - x^3}{1 - 2x^2} \right) \].

Step 6: Check Against Options

For options A, B, C, and D, the correct match for \( y \) is \( \frac{3x - x^3}{1 + 3x^2} \), matching option (B).

Key Concepts

Understanding the Inverse Tangent Addition Formula

The inverse tangent addition formula is a very useful tool when dealing with equations involving inverse tangent (_arctan_) functions. The identity to remember here is:
\[ \tan^{-1} a + \tan^{-1} b = \tan^{-1}\left( \frac{a+b}{1-ab} \right) \]
This formula is applicable when the product \(ab < 1\). What this identity does is it combines two inverse tangents into one, making it much simpler to solve equations that have several terms with inverse tangents.

• It is crucial because it allows us to take the sum of two separate angles and combine them into a single angle.
• By simplifying the expression, it becomes much easier to solve for unknown variables.
• This formula is widely used in calculus and trigonometry to simplify complex expressions into more manageable forms.

The key takeaway is that whenever you see an equation with the sum of two inverse tangents, considering the application of this addition formula can provide an immediate path to simplification.

Tangent Identity Simplification

Simplifying tangent identities is another key part of solving trigonometric equations. In our original exercise, simplification steps were used multiple times to reach the final result.
To apply the inverse tangent addition formula, simplify terms like
\[ x + \frac{2x}{1-x^2} \]This involves finding a common denominator and combining results:
\[ \frac{x(1-x^2) + 2x}{1-x^2} = \frac{x - x^3 + 2x}{1-x^2} = \frac{3x - x^3}{1-x^2} \]Furthermore, simplification may also be required in the denominator, as was done in:
\[ 1 - x \cdot \frac{2x}{1-x^2} = \frac{1 - 2x^2}{1-x^2} \]Simplicity in expressions makes solving for unknowns straightforward. It helps us handle complex mathematical problems without getting bogged down by lengthy calculations. Ultimately, practicing these methods of simplification fosters a deeper understanding of the material.

The Art of Trigonometric Equation Solving

Solving trigonometric equations, like the one in the original exercise, often involves combining several mathematical concepts: using identities, simplifying expressions, and analyzing solution options.

• Use given identities like the tangent addition formula to reduce complexity. It makes comparison with provided options easier.
• Simplify expressions correctly by handling numerators and denominators separately.
• Understand the conditions under which identities hold; for instance, ensuring \(ab < 1\) for the tangent addition formula.

When armed with trigonometric identities and a strategy for simplification, solving equations becomes an exercise in applying logic step by step. The added advantage is that it enhances problem-solving skills broadly applicable across mathematics.
Mastering these techniques can elevate a student’s understanding from just solving textbook problems to confidently tackling unfamiliar trigonometric expressions.