Problem 19 - Technical Mathematics with Calculus

Question

Prove each identity. $$\frac{1+\tan x}{1-\tan x}=\tan \left(\frac{\pi}{4}+x\right)$$

Step-by-Step Solution

Answer

\( \frac{1 + \tan x}{1 - \tan x} = \tan(\frac{\pi}{4} + x) \) identity is proven using the tangent addition formula and the value of \( \tan(\frac{\pi}{4}) \).

Step 1: Express tan(π/4 + x) using the tangent addition formula

The tangent addition formula states that \( \tan(\alpha + \beta) = \frac{\tan\alpha + \tan\beta}{1 - \tan\alpha \tan\beta} \). Applying this to \( \tan(\frac{\pi}{4} + x) \), set \( \alpha = \frac{\pi}{4} \) and \( \beta = x \), we get \( \tan(\frac{\pi}{4} + x) = \frac{\tan(\frac{\pi}{4}) + \tan(x)}{1 - \tan(\frac{\pi}{4})\tan(x)} \).

Step 2: Simplify using the known values of tan(π/4)

Since \( \tan(\frac{\pi}{4}) = 1 \), the formula simplifies to \( \frac{1 + \tan(x)}{1 - \tan(x)} \). This matches the left-hand side of the original identity.

Step 3: Conclude the proof

Since both the left-hand side and right-hand side of the equation are simplified to the same expression \( \frac{1 + \tan(x)}{1 - \tan(x)} \), we have proven the identity \( \frac{1 + \tan x}{1 - \tan x} = \tan(\frac{\pi}{4} + x) \).

Key Concepts

Tangent Addition Formula

The tangent addition formula is an essential trigonometric identity that helps in computing the tangent of the sum of two angles. According to this formula, for any two angles \( \alpha \) and \( \beta \) the tangent of their sum, \( \tan(\alpha + \beta) \) can be expressed as \( \frac{\tan\alpha + \tan\beta}{1 - \tan\alpha \tan\beta} \).

Understanding this formula is crucial for solving various trigonometric problems, including proofs like the one in the original exercise. It allows us to break down complex angles into more manageable ones. For instance, in the given problem, the angle \( \frac{\pi}{4} + x \) is dissected into the sum of \( \frac{\pi}{4} \) and \( x \) which are then easily managed using their individual tangents.

Proof of Trigonometric Identities

The process of proving trigonometric identities often involves several steps, including rearranging the equation, substituting known trigonometric values, and simplifying expressions. A successful proof shows that two different expressions are equivalent, meaning they have the same value for all valid angles. In the exercise provided, the proof starts by using the tangent addition formula and simplifies the expression by substituting the known value of \( \tan(\frac{\pi}{4}) \). These sorts of proofs are essential for understanding the relationship between various trigonometric functions and for developing the ability to tackle more complex mathematical challenges.

When proving a trigonometric identity like the one given, it's important to be methodical. Start from known identities, clearly show each step, and ensure that each transformation you make is based on established mathematical relationships. This rigorous method helps ensure that the proof is sound and understandable.

Mathematical Proofs

Mathematical proofs are logical arguments that establish the truth of a mathematical statement. They are a foundation of mathematical reasoning and require evidence that is universally accepted by the mathematical community. A proof can take various forms, such as direct proof, proof by contradiction, or induction, and its core purpose is always to show that something is an undeniable fact within the assumptions of a given mathematical system.

In our textbook example, the direct proof method is used. This involves assuming that the formula for the tangent addition is true, then through algebraic manipulation and known trigonometric values, demonstrating that the two sides of the identity are equivalent. This kind of logical reasoning is a fundamental skill in mathematics and helps students not only to verify the results but also to gain a deeper understanding of the underlying principles and how different areas of mathematics are interconnected.