Problem 32 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The identity is proven using the tangent subtraction formula.

Step 1: Understand the Identity

Our task is to prove the trigonometric identity \( \tan \left(x - \frac{\pi}{4}\right) = \frac{\tan x - 1}{\tan x + 1} \). This can be done by using the tangent subtraction formula.

Step 2: Recall the Tangent Subtraction Formula

The tangent subtraction formula is given by: \[ \tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b} \] In this identity, \( a = x \) and \( b = \frac{\pi}{4} \).

Step 3: Apply the Tangent Subtraction Formula

Substituting \( a = x \) and \( b = \frac{\pi}{4} \) into the formula: \[ \tan \left(x - \frac{\pi}{4}\right) = \frac{\tan x - \tan \frac{\pi}{4}}{1 + \tan x \tan \frac{\pi}{4}} \] Since \( \tan \frac{\pi}{4} = 1 \), this simplifies to: \[ \tan \left(x - \frac{\pi}{4}\right) = \frac{\tan x - 1}{1 + \tan x} \]

Step 4: Verify the Results

Compare the result from Step 3, \( \frac{\tan x - 1}{1 + \tan x} \), with the given right-hand side of the identity, \( \frac{\tan x - 1}{\tan x + 1} \). They are equal, confirming the identity.

Key Concepts

Tangent Subtraction Formula

When working with angles and trigonometric identities, one key formula to remember is the tangent subtraction formula. This formula is widely used to simplify expressions involving the tangent of a difference between two angles. The formula is stated as follows:

• \[ \tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b} \]

This equation helps us expand or contract expressions where two angles are subtracted, making it easier to manipulate and prove identities. In practical use, you identify the angles 'a' and 'b' and substitute them into the formula.
For instance, in the given problem, we have \( a = x \) and \( b = \frac{\pi}{4} \), which allows us to use the formula to prove that \( \tan \left(x-\frac{\pi}{4}\right) \) can be expressed as \( \frac{\tan x - 1}{\tan x + 1} \)."

Trigonometric Functions

Trigonometric functions are foundational elements in mathematics, especially in geometry and calculus. They describe the relationships between the angles and lengths of triangles. The primary trigonometric functions are sine (sin), cosine (cos), and tangent (tan). Each function has its unique characteristics, but all are interrelated through various mathematical identities. Among these, the tangent function is defined as the ratio:

• \( \tan \theta = \frac{\text{opposite side}}{\text{adjacent side}} \)

This definition helps in analyzing right-angled triangles, or angles on the unit circle.
The tangent function is periodic with a period of \( \pi \) and is undefined at odd multiples of \( \frac{\pi}{2} \). Understanding these properties helps in solving problems involving angle subtraction or addition. The tangent subtraction formula is part of these identities, helping to bridge the concepts between pure geometric interpretation and algebraic manipulation."

Angle Subtraction Identity

The angle subtraction identity is a crucial concept in trigonometry that allows the simplification of expressions involving the subtraction of two angles. These identities form the backbone of advanced trigonometry and are used in calculus, physics, and engineering.
The tangent subtraction identity specifically is:

• \[ \tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b} \]

Using this identity helps in rewriting expressions in a manageable form, facilitating easier computation or further transformation.
In the context of the exercise provided, the angle subtraction identity directly applies to transform \( \tan \left(x-\frac{\pi}{4}\right) \) into \( \frac{\tan x - 1}{\tan x + 1} \). This not only confirms the identity but reinforces how essential such identities are in proving or simplifying complex trigonometric expressions."