Problem 1 - A Complete Resource Book in Mathematics for JEE Main

Question

The equation \(2 \cos ^{2}\left(\frac{x}{2}\right) . \sin ^{2} x=x^{2}+\frac{1}{x^{2}} 0 \leq x \leq \frac{\pi}{2}\) has (A) one real solution (B) no solution (C) more than one real solution (D) none of these

Step-by-Step Solution

Answer

(B) no solution.

Step 1: Understand the Equation

The given problem deals with the trigonometric identity equation \(2 \cos ^{2}\left(\frac{x}{2}\right) \cdot \sin ^{2} x = x^{2}+\frac{1}{x^{2}}\). We need to check if there are any real solutions in the interval \([0, \pi/2]\) by analyzing the left and right side of the equation.

Step 2: Analyze the Trigonometric Side

Rewriting the left-hand side: \(2 \cos^2(\frac{x}{2}) \cdot \sin^2(x) = 2(1-\sin^2(\frac{x}{2})) \cdot \sin^2(x)\). The expression involves trigonometric functions that vary between 0 and 1.

Step 3: Investigate the Algebraic Side

For the right-hand side, analyzing \(x^2 + \frac{1}{x^2}\) for \(x > 0\), we note that this expression trends upwards quickly and has singularity issues as \(x\) approaches zero.

Step 4: Boundary and Critical Point Analysis

At \(x = 0\), the expression \(x^2 + \frac{1}{x^2}\) becomes undefined, and as \(x \to 0^+\), \(\frac{1}{x^2}\) dominates, going to infinity. At \(x = \pi/2\), \(\cos^2(\frac{\pi}{4})\) results in minimal value contributions from the trigonometric side.

Step 5: Solve for Real Solutions

Plug in possible simple values for checking feasibility within the domain \(0 \leq x \leq \pi/2\). However, we don't find any \(x\) where \(2 \cos ^{2}(\frac{x}{2}) \cdot \sin^2(x)\) matches \(x^2 + \frac{1}{x^2}\).

Step 6: Conclusion

No matches indicate no real solutions in the domain, suggesting the problem has (B) no solution as analyzing through assumed critical points and endpoints failed.

Key Concepts

Trigonometric Identities

Trigonometric identities are pivotal in solving problems involving trigonometric functions. They are equations involving trigonometric functions that hold true for any value of the variable. In trigonometric equations, these identities help simplify complex expressions and reveal hidden relationships.

For example, the identity \( \cos^2(\frac{x}{2}) = 1 - \sin^2(\frac{x}{2}) \) helps to transform and simplify the trigonometric expression in the equation given:

• \( 2 \cos^2(\frac{x}{2}) \cdot \sin^2(x) \).

By substituting the identity, the expression on the left-hand side becomes more manageable, making it easier to analyze and solve.

Using trigonometric identities strategically is essential for tackling trigonometric equations, as they can uncover solutions that might not be evident initially. Recognizing when and how to apply these identities allows for a deeper understanding of trigonometric equations.

Real Solutions

When working with trigonometric equations, particularly when determining real solutions, it's crucial to consider the domain and range of the functions involved. Real solutions are the values of \( x \) that satisfy the equation when \( x \) is restricted to a real number.

In the context of the provided exercise, we focus on the interval \([0, \frac{\pi}{2}]\). It's essential to check for solutions within this interval as trigonometric functions are commonly defined in specific ranges.

The challenge arises from comparing the behavior of the two sides of the equation:

• The left side involves trigonometric functions that oscillate within a fixed range.
• The right side is algebraic, comprising \( x^2 + \frac{1}{x^2} \), which behaves differently, with a rapid increase and singularity issues at zero.

After careful investigation, determining that no real solutions exist within the specified domain emphasizes the importance of understanding the distinct characteristics of trigonometric and algebraic expressions when searching for real solutions.

Algebraic Analysis

Algebraic analysis involves breaking down the algebraic component of an equation to understand its structure and behavior. In this case, the expression \( x^2 + \frac{1}{x^2} \) is key to the right side of the equation.

This expression has unique characteristics:

• It demonstrates a rapid increase in values, especially as \( x \) moves away from zero.
• As \( x \to 0^+ \), the term \( \frac{1}{x^2} \) dominates, leading to an asymptotic behavior where it approaches infinity.

Analyzing this helps us ascertain that, while the algebraic side grows unbounded as \( x \) nears zero, it remains continuous over the interval \((0, \pi/2]\).

This analysis highlights that the algebraic side doesn't offer any feasible matching "real" values within the specified interval to equate to the trigonometric side. Understanding these nuances ensures accurate evaluation of potential solutions and solidifies why no real solutions exist for the equation in the given interval.