Problem 21 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(\theta=\tan ^{-1} \alpha, \varphi=\tan ^{-1} b\) and \(a b=-1\) then \(\theta-\varphi=\) (A) 0 (B) \(\frac{\pi}{4}\) (C) \(\frac{\pi}{2}\) (D) none of these

Step-by-Step Solution

Answer

The correct answer is (C) \( \frac{\pi}{2} \).

Step 1: Understand the Given Information

We are given two angles, \( \theta \) and \( \varphi \), defined as \( \theta = \tan^{-1} \alpha \) and \( \varphi = \tan^{-1} b \). Additionally, the relationship \( a b = -1 \) is provided.

Step 2: Use the Tangent Difference Formula

The tangent of the difference of two angles \( \theta \) and \( \varphi \) is given by: \[ \tan (\theta - \varphi) = \frac{ \tan \theta - \tan \varphi }{ 1 + \tan \theta \tan \varphi } \].

Step 3: Substitute Values from the Given Equations

Substitute \( \tan \theta = \alpha \) and \( \tan \varphi = b \) into the formula from Step 2: \[ \tan (\theta - \varphi) = \frac{ \alpha - b }{ 1 + \alpha b } \].

Step 4: Apply the Condition \(ab = -1\)

Use the condition \( \alpha b = -1 \) in the formula: \[ 1 + \alpha b = 1 - 1 = 0 \]. Since the denominator is zero, \( \tan(\theta - \varphi) \) becomes undefined, indicating \( \theta - \varphi = \frac{\pi}{2} \).

Step 5: Determine the Right Answer Choice

According to Step 4, since \( \theta - \varphi = \frac{\pi}{2} \) when the tangent is undefined, the correct answer is (C) \( \frac{\pi}{2} \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are the reverse processes of the regular trigonometric functions, such as sine, cosine, and tangent. These functions serve an essential role when you need to find an angle given a specific trigonometric value. For example, if you know the tangent of an angle is a particular number, you use the inverse tangent, written as \( \tan^{-1} \) or \( \arctan \), to find the angle itself.
Inverse trigonometric functions have specific ranges to ensure they provide unique results:

• The range of \( \tan^{-1} \) is \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \).
• It implies that \( \tan(\theta) \) produces an angle in this range.

These functions become particularly handy when solving equations within trigonometry, like the one given in the exercise. Understanding inverse trigonometric functions is crucial as they allow for the determination of actual angle measures, which can then be used to solve further problems, especially involving identities.

Tangent Formula

The tangent formula is vital when dealing with the difference or addition of angles. Specifically, the tangent of the difference of two angles \( \theta \) and \( \varphi \) is calculated using the formula:

• \( \tan(\theta - \varphi) = \frac{ \tan \theta - \tan \varphi }{ 1 + \tan \theta \tan \varphi } \)

This formula helps relate the trigonometric values of individual angles and the resultant angle formed by their difference or sum.
In the exercise, this tangent formula helps determine the undefined nature of \( \tan(\theta - \varphi) \) due to the denominator equating to zero when \( \alpha b = -1 \). This condition highlights the critical property of tangent becoming undefined and directly leading to a specific angle value of \( \frac{\pi}{2} \). Understanding the tangent formula's use and inherent nature of undefined values informs how different angles interact and result in specific angles.

Angle Difference Identities

Trigonometric identities are equations that hold true for all angle values and provide relationships between various trigonometric functions. The angle difference identities, like the one used here, play a pivotal role in manipulating angles to find their trigonometric values.
For the tangent, this identity expresses how the tangent of an angle difference is the relationship between the tangents of the individual angles:

• \( \tan(\theta - \varphi) = \frac{ \tan \theta - \tan \varphi }{ 1 + \tan \theta \tan \varphi } \)

In the original exercise, the identity provided highlights how specific angle scenarios lead to certain angle conclusions, such as \( \theta - \varphi = \frac{\pi}{2} \), when the expression becomes undefined due to a zero denominator.
Moreover, these identities help students recognize patterns that angles follow and understand how changes in the structure of trigonometric expressions influence the values. Mastering these identities is crucial, as they allow for simplifications and solutions to more complex trigonometry problems.