Problem 45 - Algebra and Trigonometry

Question

Rewrite the expression as an algebraic expression in \(x .\) \(\cos \left(2 \tan ^{-1} x\right)\)

Step-by-Step Solution

Answer

The expression is rewritten as \( \frac{1 - x^2}{x^2 + 1} \).

Step 1: Understand Trigonometric Identity

Recognize that the expression \( \cos(2\theta) \) can be rewritten using the double angle formula identity: \( \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \). In this problem, \( \theta = \tan^{-1}(x) \).

Step 2: Use Right Triangle Relationships

Since \( \theta = \tan^{-1}(x) \), a right triangle can be constructed where the opposite side to \( \theta \) is \( x \) and the adjacent side is \( 1 \). The hypotenuse becomes \( \sqrt{x^2 + 1} \). The cosine and sine of \( \theta \) are therefore \( \cos(\theta) = \frac{1}{\sqrt{x^2 + 1}} \) and \( \sin(\theta) = \frac{x}{\sqrt{x^2 + 1}} \).

Step 3: Apply the Double Angle Formula

Substitute the expressions for \( \cos(\theta) \) and \( \sin(\theta) \) into the double angle formula: \[\cos(2\theta) = \left(\frac{1}{\sqrt{x^2 + 1}}\right)^2 - \left(\frac{x}{\sqrt{x^2 + 1}}\right)^2\]

Step 4: Simplify the Expression

Calculate \( \cos^2(\theta) \) and \( \sin^2(\theta) \):\[\cos^2(\theta) = \frac{1^2}{(\sqrt{x^2 + 1})^2} = \frac{1}{x^2 + 1} \]\[\sin^2(\theta) = \frac{x^2}{(\sqrt{x^2 + 1})^2} = \frac{x^2}{x^2 + 1} \]Substitute these back into the double angle formula:\[ \cos(2\theta) = \frac{1}{x^2 + 1} - \frac{x^2}{x^2 + 1} \]

Step 5: Finalize the Formula

Combine the fractions:\[ \cos(2\theta) = \frac{1 - x^2}{x^2 + 1} \]Thus, \( \cos\left(2\tan^{-1}x\right) = \frac{1 - x^2}{x^2 + 1} \).

Key Concepts

Double Angle Formulas

Double angle formulas are a crucial part of trigonometry that help us simplify expressions involving angles, usually to avoid dealing with complex calculations. The most commonly used double angle formulas are for sine, cosine, and tangent. In this context, we're using the cosine double angle formula:

• \( \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \)
• Alternatively, \( \cos(2\theta) \) can also be expressed as \( 2\cos^2(\theta) - 1 \) or \( 1 - 2\sin^2(\theta) \).

These variations can be helpful depending on the given problem. Here, once we identify \( \theta \) as \( \tan^{-1}(x) \), we construct the equivalent trigonometric identities using a geometric approach. By forming a right triangle, we gain straightforward access to the values of cosine and sine, which are essential for applying the double angle formula. This method simplifies our expression significantly, leading us to an algebraic expression involving \( x \).

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to find angles when trigonometric ratios are known. In the problem at hand, we deal with \( \tan^{-1}(x) \), which returns the angle \( \theta \) whose tangent is \( x \). Here's what you should remember:

• \( \tan^{-1}(x) \) is also called "arctan" and maps a ratio back to an angle, typically lying between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\).
• It plays a crucial role in converting a trigonometric expression into a form that utilizes easier algebraic expressions.

In our approach, we create a right triangle to visualize this better, where \( x \) is the side opposite the angle \( \theta \), 1 is the adjacent side, and the hypotenuse is \( \sqrt{x^2 + 1} \). This setup enables us to determine values for cosine and sine of \( \theta \), leading to the simplification of \( \cos(2\tan^{-1}(x)) \). This visualization strategy makes it much simpler, turning a potentially abstract problem into something more tangible.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators like addition or multiplication. They provide a simplified representation of a problem and are foundational in mathematics. For the given solution, we start with a trigonometric expression and aim to express it as an algebraic expression. We use:

• Trigonometric identities to convert trigonometric functions into algebraic forms.
• The relationship between trigonometric functions and right angle triangles to solve for cosine and sine of the angle provided by \( \tan^{-1}(x) \).
• Simplification skills to combine the fractions and derive the final result.

The goal is to move from something like \( \cos(2\tan^{-1}(x)) \) to \( \frac{1 - x^2}{x^2 + 1} \), a cleaner, simpler form that uses basic algebra involving \( x \). Such transformations are key in solving complex trigonometric problems without excessive computation.