Problem 31 - Algebra and Trigonometry Real Mathematics, Real People

Question

Describe the error. $$\begin{aligned} &(1+\tan x)[1+\cot (-x)]\\\ &\begin{array}{l} =(1+\tan x)(1+\cot x) \\ =1+\cot x+\tan x+\tan x \cot x \\ =1+\cot x+\tan x+1 \\ =2+\cot x+\tan x \end{array} \end{aligned}$$

Step-by-Step Solution

Answer

The incorrect part in the exercise is substitution of \( \cot(-x) \) with \( \cot x \), whereas it should be substituted with \(-\cot x \). Hence, the corrected equation is \( =\tan x - \cot x \)

Step 1: Identify Mistake

In the given expression \( (1+\tan x)[1+\cot (-x)] \), \(-x\) is used in place of \(x\) and has incorrectly been defined as \( \cot x \). They are not the same. Remember that \( \cot (-x) = -\cot x \). Thus, \(-\cot x\) should be used instead of \(\cot x\) because it is negative of \(x\).

Step 2: Correcting The Error

Now, the correct version of the mathematical computation should be: Start with the given expression \( (1+\tan x)[1+\cot(-x)] \).Now substitute \( \cot(-x) \) as \(-\cot x\):\(=(1+ \tan x)(1-\cot x) \)And continue simplification:\( =1-\cot x+\tan x-\tan x \cot x \)Notice that the error term \( \tan x \cot x \) cancels out since \( \tan x \cot x = 1 \).Thus the final corrected version of the equation should be:\( =1-\cot x+\tan x-1 \)Which simplifies to \( =\tan x - \cot x \)

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, particularly in the study of triangles and waves. In this exercise, we deal with functions like tangent (\( \tan x \)) and cotangent (\( \cot x \)).

These functions are related to each other, as \( \cot x = \frac{1}{\tan x} \) and \( \tan x = \frac{\sin x}{\cos x} \). Understanding these relationships is crucial when performing algebraic manipulations involving trigonometric identities.

Recognizing that certain trigonometric functions change sign with negative angles is important. For example, \( \tan(-x) = -\tan x \) and \( \cot(-x) = -\cot x \). This knowledge is essential for solving trigonometric problems correctly, especially when substituting values into expressions that involve negative angles.

Mathematical Errors

Mathematical errors can occur when incorrect assumptions are made about identities or when signs are not properly handled. In our original exercise, the mistake arises when translating \( \cot(-x) \) directly to \( \cot x \) without accounting for the sign change.

• The expression \( 1 + \cot(-x) \) should be evaluated as \( 1 - \cot x \) due to the property of the cotangent function with negative angles.


• Such errors can often lead to incorrect simplifications, making it crucial to keep track of these trigonometric properties.

When working through problems involving trigonometric functions, it’s essential to double-check which identities and properties are being used. Properly identifying and applying the correct values prevents mistakes, ensuring that calculations lead to the correct answers.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions using basic algebra rules and identities. This skill is particularly useful in solving trigonometric equations, as seen in the corrected steps provided in this exercise.

When simplifying trigonometric expressions, it's important to:

• Recognize valid trigonometric identities, such as \( \tan x \cot x = 1 \), which implies that multiplying tangent by cotangent simplifies to 1.


• Pay close attention to the signs of trigonometric functions to ensure terms are combined or canceled correctly.

In the corrected solution, careful algebraic manipulation allowed for the cancellation of the term \( \tan x \cot x \), leading to the correct simplified form \( \tan x - \cot x \). Building a strong foundation in algebraic techniques and recognizing common identities helps with the efficient solving of similar exercises.