Problem 1 - A Complete Resource Book in Mathematics for JEE Main 2017

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

The equation has no solution, thus option (B) is correct.

Step 1: Understanding the Problem

We need to find the number of solutions to the equation \(2 \cos^{2}\left(\frac{x}{2}\right) \sin^{2}x = x^{2} + \frac{1}{x^{2}}\) within the interval \(0 \leq x \leq \frac{\pi}{2}\).

Step 2: Simplifying the Equation

The given equation is \(2 \cos^{2}\left(\frac{x}{2}\right) \sin^{2}x = x^{2} + \frac{1}{x^{2}}\). First, simplify the left side using trigonometric identities. We have \( \cos^{2}\left(\frac{x}{2}\right) = \frac{1 + \cos x}{2}\) and \( \sin^{2}x = 1 - \cos^2 x\), but lets solve using another approach.

Step 3: Substituting Known Values

Since this is a trigonometric and algebraic hybrid equation, check for solutions by testing simple angles. Start with \(x = 0\): \(2 \cos^{2}(0) \sin^{2}(0) = 0^2 + \frac{1}{0^2}\) which is not defined.

Step 4: Checking Specific Angles

Try \(x = \frac{\pi}{2}\): we have \(2 \cos^{2}\left(\frac{\pi}{4}\right) \cdot \sin^{2}\left(\frac{\pi}{2}\right)\) which is \(\frac{1}{2}\cdot 1 = 1\), but \( \left(\frac{\pi}{2}\right)^2 + \frac{1}{\left(\frac{\pi}{2}\right)^2}\) which does not equal 1.

Step 5: Analyzing Function Behavior

Analyze the behavior of the left and right sides of the equation. The left side is always non-negative and maximum is 1 since \(\sin^2 x\) is part of it. Similarly, the right side varies with \(x\). With \(x >1, \frac{1}{x^2}\) diminishes but \(x^2\) increases.

Step 6: Comparing at Boundary Values

It's important to check the equation behavior around boundary and critical points. Let's test other values near zero: for very small \(x\), left = 0 but right \(\approx \frac{1}{x^2}\) large.

Step 7: Conclusion from Analysis

The equation fails to balance for any value within the bounds including testing known boundary values as they are contradictory. The behavior of each side demonstrates no equality occurs.

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental tools in solving trigonometric equations. These identities help simplify complex expressions by expressing one trigonometric function in terms of another. Here are a few critical identities that you should keep in mind when dealing with trigonometric equations like the one in our exercise:

• The identity for cosine squared: \[\cos^{2}\left(\frac{x}{2}\right) = \frac{1 + \cos x}{2}\]This identity allows us to express \(\cos^{2}\left(\frac{x}{2}\right)\) in terms of \(\cos x\).
• The identity for sine squared: \[\sin^{2}x = 1 - \cos^2 x\]This helps reform \(\sin^{2}x\) if needed, using \(\cos x\).

When tackling an equation involving these identities, the goal is to simplistically rewrite parts of the equation for easier manipulation or comparison. In the equation from the exercise, knowing when and how to apply these identities correctly can make analyzing and solving the equation significantly easier.

Trigonometric Equations

Trigonometric equations often require both algebraic and trigonometric techniques to solve. Typically, these equations involve trigonometric functions such as sine, cosine, or tangent equaling some other algebraic expression. Here's how the approach generally works:

• Recognize the elements in the equation that can be simplified using identities as discussed earlier.
• After simplification, consider substituting known values for \(x\), especially within a given interval. For example:
  Trying simple or boundary values can often hint at possible solutions or contradictions.
• Analyze any patterns or symmetries in the equation given the properties of trigonometric functions in the specific range.

After these initial strategies, further steps may involve applying numerical or graphical methods if the algebraic approach becomes cumbersome. In our problem, analyzing specific angles, such as \(x = 0\) and \(x = \frac{\pi}{2}\), along with analyzing behavior of trigonometric and algebraic components was key to determining if solutions exist.

Solution Analysis

Analyzing the solution of a trigonometric equation requires understanding both the mathematical rigor and the conceptual framework. This involves:

• Breaking down the equation with boundary values to ensure solutions meet logical conditions.
• Examining each component's behavior, such as understanding how \(\sin^2 x\) affects values based on its range \([0,1]\).
• Comparing the consistency of behavior on both sides of the equation. In our problem, we noted that one side can reach a maximum of 1 while the other side involving \(x^2 + \frac{1}{x^2}\) can grow significantly larger.

This thorough comparison helps in deducing that the equation inequalities do not allow a valid solution in the specified range, leading to a conclusive understanding that the equation has no solution within \(0 \leq x \leq \frac{\pi}{2}\). Solution analysis is not just about numerical accuracy but also about logical deductions from function behaviors.