Problem 45 - Algebra and Trigonometry

Question

Write the given expression in terms of x and y only. $$ \sin \left(\tan ^{-1} x-\tan ^{-1} y\right) $$

Step-by-Step Solution

Answer

\( \sin(\tan^{-1} x - \tan^{-1} y) = \frac{x - y}{\sqrt{(x^2 + 1)(y^2 + 1)}} \).

Step 1: Understand the Formula

We are given the expression \( \sin(\tan^{-1} x - \tan^{-1} y) \). The task is to express this in terms of \( x \) and \( y \).

Step 2: Use the Angle Difference Identity

Recall the angle difference identity for sine: \( \sin(A - B) = \sin A \cos B - \cos A \sin B \). We will apply this identity to the angles \( A = \tan^{-1} x \) and \( B = \tan^{-1} y \).

Step 3: Express \( \sin A \) and \( \cos A \) in Terms of \( x \)

For \( A = \tan^{-1} x \), we can imagine a right triangle where the opposite side is \( x \) and the adjacent side is 1, making the hypotenuse \( \sqrt{x^2 + 1} \). Thus, \( \sin A = \frac{x}{\sqrt{x^2 + 1}} \) and \( \cos A = \frac{1}{\sqrt{x^2 + 1}} \).

Step 4: Express \( \sin B \) and \( \cos B \) in Terms of \( y \)

For \( B = \tan^{-1} y \), set up a triangle with opposite side \( y \) and adjacent side 1, with hypotenuse \( \sqrt{y^2 + 1} \). Then \( \sin B = \frac{y}{\sqrt{y^2 + 1}} \) and \( \cos B = \frac{1}{\sqrt{y^2 + 1}} \).

Step 5: Substitute Back into the Sine Difference Identity

Substitute \( \sin A, \cos A, \sin B, \) and \( \cos B \) into the identity:\[ \sin(\tan^{-1} x - \tan^{-1} y) = \frac{x}{\sqrt{x^2+1}} \cdot \frac{1}{\sqrt{y^2+1}} - \frac{1}{\sqrt{x^2+1}} \cdot \frac{y}{\sqrt{y^2+1}} \]

Step 6: Simplify the Expression

Simplify the expression by combining the fractions over a common denominator: \[ \sin(\tan^{-1} x - \tan^{-1} y) = \frac{x - y}{\sqrt{(x^2 + 1)(y^2 + 1)}} \]. This is the expression in terms of \( x \) and \( y \).

Key Concepts

Angle Difference Identity

The angle difference identity is a significant mathematical tool that helps simplify and solve trigonometric problems. When dealing with triangles or periodic functions like sine or cosine, knowing how to work with angles that are either added or subtracted is crucial. The specific identity for sine that is used in our problem is \( \sin(A - B) = \sin A \cos B - \cos A \sin B \). This formula allows us to break down the sine of a difference into more manageable parts by expressing it in terms of the sines and cosines of the individual angles.

In our context, the angles \( A \) and \( B \) are inverse tangents, namely \( A = \tan^{-1} x \) and \( B = \tan^{-1} y \). By substituting these into the identity, the problem can be simplified to basic algebraic functions using the characteristics of sine and cosine. The magic here is in converting complex angle expressions into something simpler using basic trigonometric ratios from a right triangle model, enabling a clearer path to solve the expression in terms of \( x \) and \( y \).

Inverse Trigonometric Functions

Inverse trigonometric functions like \( \tan^{-1}, \sin^{-1}, \) and \( \cos^{-1} \) are essential for finding angles when we know some trigonometric ratios. For instance, \( \tan^{-1} x \) gives us the angle whose tangent is \( x \). Here, these functions are pivotal because they allow us to transform a given ratio into a known angle measure.

To better understand \( \tan^{-1} x \), you can visualize a right triangle where the opposite side is \( x \) and the adjacent side is \( 1 \). The hypotenuse would then be calculated as \( \sqrt{x^2 + 1} \), making it easy to derive \( \sin A = \frac{x}{\sqrt{x^2 + 1}} \) and \( \cos A = \frac{1}{\sqrt{x^2 + 1}} \) using basic trigonometry.

This approach is also applied to \( \tan^{-1} y \) and helps break down a complex trigonometric expression into straightforward components. It allows one to rewrite trigonometric expressions in a way that is easier to manipulate algebraically. Understanding and using these inverse functions effectively transforms intimidating trigonometric expressions into simpler, solvable terms.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions can initially appear daunting, yet through systematic application of trigonometric identities and algebraic rules, it becomes manageable. The goal is to convert the presented expression into a simpler form, usually involving basic operations such as addition, subtraction, multiplication, or division, while eliminating any unnecessary complexity.

In solving \( \sin(\tan^{-1} x - \tan^{-1} y) \), we utilize identities and inverses: starting from the angle difference identity, substituting the sine and cosine for each inverse tangent in terms of \( x \) and \( y \), and reducing excess complexity. At the end, the objective is to achieve an expression like \( \frac{x - y}{\sqrt{(x^2 + 1)(y^2 + 1)}} \), where the result is both compact and simplified fully in terms of the variables \( x \) and \( y \).

To get to this point, it’s crucial to remember:

• Break down expressions using known identities and functions.
• Look for common factors and simplify fractions when possible.
• Always check that your final expression makes sense in context, making it clear and concise.

This process involves thoughtful application of mathematical reasoning and a bit of creativity!