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Magazine Chapter 19: Problem 3
Show that $$ \tan \left(x+\frac{\pi}{4}\right) \tan \left(x-\frac{\pi}{4}\right)=-1 $$
Short Answer
Expert verified
The expression simplifies to -1 using tangent addition and subtraction formulas.
Step by step solution
01
Use the addition (subtraction) formula for tangent
We start by using the tangent addition and subtraction formulas. These are:\[ \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \]and\[ \tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b} \]We are specifically interested in \( \tan\left(x + \frac{\pi}{4}\right) \) and \( \tan\left(x - \frac{\pi}{4}\right) \). Substitute \( a = x \) and \( b = \frac{\pi}{4} \).
02
Substitute \( b = \frac{\pi}{4} \) into tangent formulas
Now substitute \( b = \frac{\pi}{4} \) into the addition and subtraction formulas. We know that \( \tan\left(\frac{\pi}{4}\right) = 1 \). So the formulas become:\[ \tan\left(x + \frac{\pi}{4}\right) = \frac{\tan x + 1}{1 - \tan x} \]and\[ \tan\left(x - \frac{\pi}{4}\right) = \frac{\tan x - 1}{1 + \tan x} \].
03
Multiply both tangent expressions
To find \( \tan\left(x + \frac{\pi}{4}\right) \tan\left(x - \frac{\pi}{4}\right) \), multiply the two expressions:\[ \frac{\tan x + 1}{1 - \tan x} \times \frac{\tan x - 1}{1 + \tan x} \].
04
Simplify the product of the fractions
Simplify by multiplying numerators and denominators:Numerator: \((\tan x + 1)(\tan x - 1) = \tan^2 x - 1\)Denominator:\((1 - \tan x)(1 + \tan x) = 1 - \tan^2 x\). Thus, \[ \tan\left(x+\frac{\pi}{4}\right) \tan\left(x-\frac{\pi}{4}\right) = \frac{\tan^2 x - 1}{1 - \tan^2 x} \].
05
Recognize the identity and conclude
Notice that the expression \( \frac{\tan^2 x - 1}{1 - \tan^2 x} \) simplifies to \(-1\) because both the numerator and the denominator are negative of each other. Thus,\[ \tan\left(x+\frac{\pi}{4}\right) \tan\left(x-\frac{\pi}{4}\right) = -1 \] is verified.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding the Tangent Addition Formula
The tangent addition formula is a cornerstone in trigonometry that helps us find the tangent of a sum of two angles. In mathematical terms, it is written as:
Let's consider the expression \( \tan(x + \frac{\pi}{4}) \). Substituting \( a = x \) and \( b = \frac{\pi}{4} \) into the addition formula:
- \( \tan(a+b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \)
Let's consider the expression \( \tan(x + \frac{\pi}{4}) \). Substituting \( a = x \) and \( b = \frac{\pi}{4} \) into the addition formula:
- \( \tan(x + \frac{\pi}{4}) = \frac{\tan x + 1}{1 - \tan x} \)
Exploring Angle Subtraction in Tangent
Angle subtraction is equally important when dealing with trigonometric identities. The formula for the tangent of a subtraction of two angles is given by:
When we apply it to \( \tan(x - \frac{\pi}{4}) \), substituting \( a = x \) and \( b = \frac{\pi}{4} \), we get:
- \( \tan(a-b) = \frac{\tan a - \tan b}{1 + \tan a \tan b} \)
When we apply it to \( \tan(x - \frac{\pi}{4}) \), substituting \( a = x \) and \( b = \frac{\pi}{4} \), we get:
- \( \tan(x - \frac{\pi}{4}) = \frac{\tan x - 1}{1 + \tan x} \)
Mastering Trigonometric Simplification Techniques
Trigonometric simplification involves using identities and formulas to make expressions easier to work with. In the case of our problem, we needed to simplify:
Recognizing these patterns and identities is an excellent way for students to master trigonometric simplification, and it fosters a more profound understanding of how different trigonometric entities relate to one another.
- \[ \frac{\tan x + 1}{1 - \tan x} \times \frac{\tan x - 1}{1 + \tan x} \]
- Numerator: \((\tan x + 1)(\tan x - 1) = \tan^2 x - 1\)
- Denominator: \((1 - \tan x)(1 + \tan x) = 1 - \tan^2 x\)
Recognizing these patterns and identities is an excellent way for students to master trigonometric simplification, and it fosters a more profound understanding of how different trigonometric entities relate to one another.
