Problem 72 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The minimum value of \(a \tan ^{2} x+b \cot ^{2} x\) equals the maximum value of \(a \sin ^{2} \theta+b \cos ^{2} \theta\) where \(a>b>0\), when (A) \(a=b\) (B) \(a=2 b\) (C) \(a=3 b\) (D) \(a=4 b\)

Step-by-Step Solution

Answer

(D) \( a = 4b \)

Step 1: Understanding the Expression

The given expression is \( a \tan^2 x + b \cot^2 x \). This expression can be rewritten using \( \tan x = \frac{\sin x}{\cos x} \) and \( \cot x = \frac{\cos x}{\sin x} \). Thus, it becomes \( \frac{a \sin^2 x}{\cos^2 x} + \frac{b \cos^2 x}{\sin^2 x} \). To find the minimum value, we consider substituting suitable values of \( x \).

Step 2: Using AM-GM Inequality

Apply the AM-GM inequality: \[ a \tan^2 x + b \cot^2 x \geq 2\sqrt{a \tan^2 x \cdot b \cot^2 x} \] which simplifies to \( 2 \sqrt{ab} \). Equality holds when \( a \tan^2 x = b \cot^2 x \), or equivalently \( a \sin^4 x = b \cos^4 x \).

Step 3: Solving for x

Set \( a \sin^4 x = b \cos^4 x \). This implies \( \sin^4 x / \cos^4 x = b/a \), so \( (\sin^2 x / \cos^2 x)^2 = b/a \). Therefore, \( \tan^2 x = \sqrt{b/a} \).

Step 4: Expression for Maximum Value

Consider \( a \sin^2 \theta+b \cos^2 \theta = f(\theta) \) and note that it's a linear combination of squares of trigonometric functions. This variation suggests using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \). Then, \( f(\theta) = a (1) + (b-a) \cos^2 \theta = a + (b-a) \cos^2 \theta \).

Step 5: Maximizing f(theta)

To find the maximum value of \( f(\theta) \), maximize \( (b-a) \cos^2 \theta \) since \( (b-a) < 0 \) makes it a decreasing function. Since \( \cos^2 \theta \) ranges between 0 and 1, and max happens when \( \theta = 0 \) or multiple of \( \pi \), \( f(\theta) = b \).

Step 6: Equating Min and Max Values

The problem states the minimum value of \( a \tan^2 x + b \cot^2 x \) equals the maximum of \( a \sin^2 \theta + b \cos^2 \theta \), thus \( 2\sqrt{ab} = b \). Solving \( 2\sqrt{ab} = b \) gives \( 2\sqrt{ab}/b = a \). Simplifying, \( 2\sqrt{a/b} = 1 \) gives \( a = 4b \).

Step 7: Selecting the Correct Option

The calculations show that when \( a = 4b \), the equality condition is satisfied. Hence, option (D) is the correct answer.

Key Concepts

AM-GM Inequality

The AM-GM inequality, short for Arithmetic Mean-Geometric Mean inequality, is a fundamental concept in mathematics that asserts the arithmetic mean of non-negative numbers is always greater than or equal to their geometric mean. This is a powerful tool in optimization and proofs, especially in handling expressions involving squares and products.
In the context of the problem, we applied the AM-GM inequality to the expression \( a \tan^2 x + b \cot^2 x \). By transforming trigonometric ratios into a form suitable for inequality application, we derived that:

• \( a \tan^2 x + b \cot^2 x \geq 2\sqrt{a \tan^2 x \cdot b \cot^2 x} \).
• This inequality holds equality when \( a \tan^2 x = b \cot^2 x \), or equivalently, when the terms inside the square root balance each other perfectly.

Understanding this application helps in finding not just minimum values but also establishing relationships between distinct mathematical expressions.

Minimum and Maximum Values

Finding minimum and maximum values, especially in trigonometric expressions, is critical in solving a variety of mathematical problems. This involves analyzing expressions and determining their extreme values within allowable parameter ranges.
For the expression \( a \tan^2 x + b \cot^2 x \), the aim was to find the minimum value by balancing the terms using AM-GM inequality. The expression achieves its minimum when the terms are equalized to ensure equality condition, providing the minimum value as \( 2\sqrt{ab} \).
On the other hand, the expression \( a \sin^2 \theta + b \cos^2 \theta \), is explored for maximum potential value considering the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \). Utilizing trigonometric identities and substituting appropriate angles, the maximum value is derived when \( \cos^2 \theta = 1 \), hence reaching a value of \( b \). Understanding how to manipulate these expressions to reveal their extremes is fundamental in various mathematical fields.

Trigonometric Equations

Trigonometric equations involve finding angles that satisfy a certain condition involving trigonometric functions. These equations are not only foundational in trigonometry but are also widely used in calculus and physics.
In solving the problem at hand, balancing the trigonometric expressions required setting equations such as \( a \sin^4 x = b \cos^4 x \) which lead to the relationship \( \tan^2 x = \sqrt{b/a} \). This transformation helps in visualizing how angle measures relate to trigonometric identities and their squares.
Similarly, the adjustment of the expression \( a \sin^2 \theta + b \cos^2 \theta \) required setting boundary conditions like maximizing \( \cos^2 \theta \), showing the practical application of trigonometric equations in deriving solutions concerning geometric intensity and periodic nature. Being fluent in these transformations and manipulations opens up a multitude of solution pathways in mathematical problems.