Problem 8 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

The smallest and the largest values of \(\tan ^{-1}\left(\frac{1-x}{1+x}\right), 0 \leq x \leq 1\) are (a) \(0, \pi\) (b) \(0, \frac{\pi}{4}\) (c) \(-\frac{\pi}{4}, \frac{\pi}{4}\) (d) \(\frac{\pi}{4}, \frac{\pi}{2}\)

Step-by-Step Solution

Answer

The smallest and largest values of \( \tan^{-1}\left(\frac{1-x}{1+x}\right) \), for \( 0 \leq x \leq 1 \), are 0 and \( \frac{\pi}{4} \) respectively. Hence, the correct option is (b) \( 0, \frac{\pi}{4} \).

Step 1: Simplify the Argument

The argument of the inverse tangent function \( \frac{1-x}{1+x} \) can be recognized as the right-hand side of the identity \( \tan(\frac{\pi}{4} - a) = \frac{1 - \tan(a)}{1 + \tan(a)} \). For this equality to hold, we set \( a = \tan^{-1}(x) \). Therefore, the original expression can be simplified as \( \tan^{-1}\left(\frac{1-x}{1+x}\right) = \tan^{-1}\left(\tan\left(\frac{\pi}{4} - \tan^{-1}(x)\right)\right) \). This simplifies to \( \tan^{-1}\left(\tan\left(\frac{\pi}{4} - \tan^{-1}(x)\right)\right) = \frac{\pi}{4} - \tan^{-1}(x) \).

Step 2: Find the Maximum and Minimum Values

Next, we find the maximum and minimum values for the simplified expression \( \frac{\pi}{4} - \tan^{-1}(x) \) given the range \( 0 \leq x \leq 1 \). For \( x = 0 \), \( \tan^{-1}(0) = 0 \), hence the expression becomes \( \frac{\pi}{4} - 0 = \frac{\pi}{4} \). For \( x = 1 \), \( \tan^{-1}(1) = \frac{\pi}{4} \), hence the expression becomes \( \frac{\pi}{4} - \frac{\pi}{4} = 0 \). Thus, the smallest and largest values are 0 and \( \frac{\pi}{4} \) respectively.

Key Concepts

Tan Inverse Properties

Inverse trigonometric functions allow us to work with angles when we have the value of a trigonometric ratio. The inverse tangent, denoted as \( \tan^{-1} \) or sometimes \( \arctan \), is particularly useful in a range of mathematical problems. Understanding the properties of the inverse tangent can help simplify complex expressions.

Here are some key properties:

• The range of \( \tan^{-1} \) is \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \) (exclusive of the end values), which means it always returns an angle within this interval.
• If \( \tan(\theta) = x \), then \( \theta = \tan^{-1}(x) \), assuming \( \theta \) lies in the range stated above.
• \( \tan^{-1} \) has the property of odd symmetry, i.e., \( \tan^{-1}(-x) = -\tan^{-1}(x) \), which is useful in various analytical and graphical interpretations.

Relating to our exercise, we can use the identity \( \tan(\frac{\pi}{4} - a) = \frac{1 - \tan(a)}{1 + \tan(a)} \) in conjunction with these properties to simplify the expression and find the solution.

Finding Maximum and Minimum Values in Trigonometry

Trigonometric functions oscillate between specific maximum and minimum values. To determine these extremities in trigonometric expressions, we often use calculus or well-known properties of trigonometric functions and their inverses.

For example, the range of \( \tan^{-1} \) tells us that it can produce values between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \), and in the context of the given exercise, we specifically look at its behaviour between 0 and 1. By analyzing the behaviour of the function at these endpoints, we can establish the smallest and largest values of the expression. This conceptual approach is at the core of solving trigonometric maximum and minimum value problems and is directly applied in the given solution where the inverse tangent is evaluated at the endpoints of the interval.

Moreover, in more complex scenarios, finding maxima and minima can involve taking the derivative of the function and solving for when the derivative equals zero (indicating potential extremities), a foundational principle in calculus.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. These identities provide powerful tools for simplifying trigonometric expressions, solving equations, and even proving other mathematical properties.

Some commonly used trigonometric identities include:

• The Pythagorean identities, such as \( \sin^2(\theta) + \cos^2(\theta) = 1 \)
• Angle sum and difference identities, such as \( \sin(\theta \pm \phi) = \sin(\theta)\cos(\phi) \pm \cos(\theta)\sin(\phi) \)
• Double angle identities, like \( \sin(2\theta) = 2\sin(\theta)\cos(\theta) \)

These identities can transform the way we handle trigonometric expressions, often reducing them to more manageable forms. In the context of our exercise, an angle difference identity is used to reframe the given \( \tan^{-1} \) expression into one that can be easily evaluated for the maximum and minimum values.