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Chapter 2: Problem 206

Prove that: If \(\tan ^{-1} x+\tan ^{-1} y=\frac{\pi}{4}\), then prove that \(x+y+x y=1\).

Short Answer

Expert verified
The given statement is true. After applying the sum identity of inverse tangent, the equation simplifies to \(x+y+xy=1\), which is the required proof.

Step by step solution

01

Apply the Identity

According to the inverse tangent sum identity, \(\tan ^{-1} x + \tan ^{-1} y = \tan ^{-1} \left(\frac{x + y}{1 - x y}\right)\) when \(x y < 1\). So substitute \(\tan ^{-1} x+\tan ^{-1} y\) with \(\frac{\pi}{4}\). It results in, \(\tan\left(\frac{\pi}{4}\right) = \frac{x + y}{1 - x y}\).
02

Simplify The Relation

\(\tan\left(\frac{\pi}{4}\right)\) gives 1. So, the relation simplifies down to \(1 = \frac{x + y}{1 - x y}\).
03

Solve The Equation

To solve this equation for \(x + y + x y = 1\), we start by simplifying the rational expression by multiplying each side by \((1 - x y)\), where our equation results in \(x + y + x y = 1\).
04

Final Step

We already have \(x + y + x y = 1\). So, we have proved the given statement.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Trigonometric Identities
Trigonometric identities are fundamental tools in solving trigonometry problems. They represent relationships between the angles and sides of a triangle or relationships between trigonometric functions. The identity relevant to our exercise is the sum of inverse tangents identity. It states that if \(a\) and \(b\) are two numbers for which \(ab < 1\), then \(\tan^{-1}a + \tan^{-1}b = \tan^{-1}\left(\frac{a + b}{1 - ab}\right)\). This identity allows us to combine two inverse tangent expressions into a single expression, which often simplifies calculations or proofs.
Inverse Trigonometric Functions
Inverse trigonometric functions, such as \(\tan^{-1}\), are used to determine angles when the values of the trigonometric functions are known. In our exercise, we have the inverse tangent, also known as arctangent, tasked with finding the angle whose tangent is a given number. The sum property of inverse trigonometric functions is especially useful when dealing with problems involving multiple angles or in finding ranges of possible values. When working with these functions, it's crucial to consider their domains and ranges to avoid ambiguous results.
IIT JEE Trigonometry Problems
IIT JEE, a prestigious engineering entrance examination in India, often features challenging trigonometry problems. These problems require a strong grasp of trigonometric identities, inverse functions, and their properties. For example, combining inverse trigonometric functions as seen in our exercise is a common task in IIT JEE trigonometry questions. When preparing for such exams, mastering these identities and understanding their application is vital. Students should focus on recognizing which identity to apply and learn to manipulate expressions to match these known identities.
Differential Calculus
Differential calculus is a branch of mathematics concerned with the study of rates at which quantities change. It's closely tied to the study of trigonometry through derivatives of trigonometric functions. While our exercise does not involve calculus directly, understanding the relationship between trigonometry and calculus can enhance the problem-solving toolkit for students. For example, knowing how to derive trigonometric functions can help understand their behavior and properties, which indirectly support solving trigonometric identities and equations.

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