Problem 40 - Precalculus: Functions and Graphs

Question

Exer. \(37-46:\) Verify the identity. $$\tan \left(x-\frac{\pi}{4}\right)=\frac{\tan x-1}{\tan x+1}$$

Step-by-Step Solution

Answer

The expression is verified; both sides are equal.

Step 1: Understand the Identity Components

In the problem, we are asked to verify the identity \( \tan\left(x - \frac{\pi}{4}\right) = \frac{\tan x - 1}{\tan x + 1} \). This means we want to make sure both sides are equivalent. We'll start by recognizing the formula for \( \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \).

Step 2: Apply Identity for Tangent Difference

Let's apply the identity \( \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \) using \( A = x \) and \( B = \frac{\pi}{4} \). Thus, we have: \[ \tan\left(x - \frac{\pi}{4}\right) = \frac{\tan x - \tan\frac{\pi}{4}}{1 + \tan x \tan\frac{\pi}{4}} \].

Step 3: Substitute Known Values

We know that \( \tan\frac{\pi}{4} = 1 \). Substitute this value into the equation: \[ \tan\left(x - \frac{\pi}{4}\right) = \frac{\tan x - 1}{1 + \tan x \cdot 1} \]. Simplifying further gives us: \[ \tan\left(x - \frac{\pi}{4}\right) = \frac{\tan x - 1}{1 + \tan x} \].

Step 4: Compare Both Sides

Now compare the simplified expression with the right-hand side of the original identity. The expression \( \frac{\tan x - 1}{1 + \tan x} \) matches exactly with \( \frac{\tan x - 1}{\tan x + 1} \).

Step 5: Conclude the Verification

Since the left-hand side, \( \tan\left(x - \frac{\pi}{4}\right) \), has been shown to equal the right-hand side, \( \frac{\tan x - 1}{\tan x + 1} \), for all \( x \) where the tangent is defined, the identity is verified.

Key Concepts

Tangent Function

The tangent function is a fundamental trigonometric function, expressed as \( \tan(x) \). It is the ratio of the sine and cosine functions: \( \tan(x) = \frac{\sin(x)}{\cos(x)} \).
This can be particularly useful because it relates directly to the angle of a right triangle:

• For any angle \( x \), \( \tan(x) \) represents the ratio between the length of the side opposite the angle and the side adjacent to the angle.
• As with sine and cosine, tangent is a periodic function with period \( \pi \). This means it repeats its values every \( \pi \) radians.

Additionally, unlike sine and cosine that have maximum and minimum values of 1 and -1, tangent can vary from negative to positive infinity.
Thus, understanding the tangent function is pivotal when solving problems involving angles and their differences.

Angle Subtraction Formula

A key concept in this exercise is the angle subtraction formula for tangent.
The formula is \( \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \). This formula allows us to break down the tangent of the difference of two angles into a form that involves the tangents of the individual angles.
Here's why this is useful:

• It simplifies the process of finding the tangent for angle differences without needing to calculate sine and cosine separately.
• It is especially handy for verification of identities like in the given exercise.
• By using substitution, such as \( B = \frac{\pi}{4} \), we can simplify complex expressions to verify identities or solve equations.

In this exercise, we used \( B = \frac{\pi}{4} \), where \( \tan \left(\frac{\pi}{4}\right) = 1 \), simplifying our expression effectively.

Verifying Identities

Verifying trigonometric identities requires showing that two expressions are indeed equivalent for all values they are defined. This often involves rewriting one or both sides using known trigonometric identities so they match.
For this exercise, we took the steps:

• Recognize the identity to be verified and identify the associated known identities, such as the angle subtraction formula.
• Substitute the values or identities into the given expression.
• Simplify both sides and compare them to ensure they are equivalent.

Here, through the use of the tangent angle subtraction formula, we transformed \( \tan(x - \frac{\pi}{4}) \) into an expression that matched the right-hand side.
When each side was simplified and found to be identical, the identity was verified, as they hold true for all angles \( x \) where tangent is defined. This process not only involves numerical substitutions but also a good grasp of underlying trigonometric principles.