Problem 34 - Algebra and Trigonometry Real Mathematics, Real People

Question

Fill in the missing step(s). $$\begin{aligned} \frac{\tan x-\cot x}{\tan x+\cot x} &=\frac{\frac{\sin x}{\cos x}-\frac{\cos x}{\sin x}}{\frac{\sin x}{\cos x}+\frac{\cos x}{\sin x}} \\ &= \text { _____} \\ &=\frac{\sin ^{2} x-\cos ^{2} x}{1} \\ &=\sin ^{2} x-\cos ^{2} x \\ &= \text { _____} \\ &=1-2 \cos ^{2} x \end{aligned}$$

Step-by-Step Solution

Answer

The missing steps are \(\frac{\sin^2x - \cos^2x}{\sin x \cos x + \cos x \sin x}\) and \( 2\cos^2x - 1 = -2\cos^2 x + 1 \)

Step 1: Substitute the given values

Replace the given trigonometric terms \(\tan{x}\) and \(\cot{x}\) with \(\frac{\sin x}{\cos x}\) and \(\frac{\cos x}{\sin x}\) respectively. The equation gets transformed to \(\frac{\frac{\sin x}{\cos x}-\frac{\cos x}{\sin x}}{\frac{\sin x}{\cos x}+\frac{\cos x}{\sin x}}\)

Step 2: Find common denominators in numerator and denominator

Express the given expression with common denominators. This results in \(\frac{\sin^2x - \cos^2x}{\sin x \cos x + \cos x \sin x}\). This step fills the first gap

Step 3: Simplification

On simplification, the denominator turns out to be \(\sin x \cos x + \cos x \sin x\) is equivalent to \(1\), hence the expression becomes \(\sin^2 x - \cos^2 x\)

Step 4: Using a trigonometric identity

By using the identity \(\sin^2 x = 1 - \cos^2 x\), the expression becomes \(2\cos^2 x - 1\) which is simplified as \(-2\cos^2 x + 1\). This fills the second gap

Step 5: Final simplification

Noting that the expression we got in step 4 is just a reversal of the expression we want, the expression thus changes to \(1 - 2\cos^2 x\)

Key Concepts

Simplification

In mathematics, simplification involves transforming an expression into its simplest form. This can often make it easier to analyze and work with. When simplifying complex expressions, especially in trigonometry, it is crucial to use known identities and algebraic manipulations to break down each part of the expression.

For instance, in the exercise

• The transformation from \(\frac{\tan x - \cot x}{\tan x + \cot x}\) to \(\frac{\sin x}{\cos x} - \frac{\cos x}{\sin x} \) was achieved by using the definitions of the tangent and cotangent functions.
• Further simplification involved finding a common denominator for the complex fraction, which allowed the numerator and denominator to combine more neatly.
• Ultimately, these simplifications lead to an expression of \(\sin^2 x - \cos^2 x \), which was subsequently further simplified using trigonometric identities.

Through these steps, each transformation achieves a simpler expression, eventually leading to the final result of \(1 - 2\cos^2 x\). This outcome is much more straightforward than the initial complex fraction. Learning to simplify expressions is a fundamental skill in mathematics that supports deeper understanding and problem-solving.

Trigonometric Functions

Trigonometric functions are the building blocks of many mathematical concepts, especially in the field of trigonometry. These functions include sine, cosine, tangent, cotangent, secant, and cosecant. Understanding how these functions relate and interact with each other is essential for solving trigonometric expressions.

In this exercise:

• The tangent (\(\tan x = \frac{\sin x}{\cos x}\))and cotangent (\(\cot x = \frac{\cos x}{\sin x}\)) functions were transformed into expressions involving sine and cosine.
• This transformation sets the stage for using identities and simplifications to reduce the complexity of the overall equation.
• Recognizing and applying reciprocal identities, such as \(\tan(x) = \frac{1}{\cot(x)} \) and \(\cot(x) = \frac{1}{\tan(x)} \), also ensures the manipulation steps are valid.

Mastering these functions can allow for better manipulation and simplification of equations, leading to swift solutions. These trigonometric identities and manipulations are daily tools for calculus, physics, and engineering problems, emphasizing their importance in various fields.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and restructuring equations to highlight certain desired features or simplify a solution. Understanding how to manipulate algebraically can solve equations faster and with greater insight.

In this specific task, one crucial type of algebraic manipulation was finding common denominators. This step is often used to simplify fractions by rewriting them so they have the same denominators, making the arithmetic more straightforward.

• In the problem, the terms \(\frac{\sin x}{\cos x} - \frac{\cos x}{\sin x} \) were initially simplified by taking common denominators of \(\sin x \cos x\), combining them into a single fraction of \(\frac{\sin^2 x - \cos^2 x}{\sin x \cos x + \cos x \sin x}\).
• This allowed for the further simplification to \(\frac{\sin^2 x - \cos^2 x}{1}\), thanks to recognizing that \(\sin x \cos x + \cos x \sin x = 1\).

These algebraic manipulations are key in shifting complex fractions into more manageable forms. Often these transformations hinge on recognizing equivalent expressions and identities. Consequently, proficiency in algebraic manipulation not only simplifies calculations but also deepens understanding of mathematical concepts.