Problem 22 - Precalculus with Calculus Previews

Question

Reduce the given expression to a single trigonometric function. $$ \left(\sin ^{2} x-1\right)\left(\cot ^{2} x+1\right) $$

Step-by-Step Solution

Answer

The expression reduces to \(-\cot^2 x\).

Step 1: Apply Pythagorean Identity

Recognize that \( \sin^2 x - 1 = -\cos^2 x \) by using the identity \( \sin^2 x + \cos^2 x = 1 \).

Step 2: Use Reciprocal Identity

Know that \( \cot^2 x + 1 = \csc^2 x \) by using the identity \( \csc^2 x = 1 + \cot^2 x \).

Step 3: Substitute Rewritten Identities

Replace the parts in the expression with their identities: \(-\cos^2 x \cdot \csc^2 x \).

Step 4: Simplify Using Reciprocal Definitions

Recall that \( \csc^2 x = \frac{1}{\sin^2 x} \), hence the expression becomes \(-\cos^2 x \cdot \frac{1}{\sin^2 x} = -\frac{\cos^2 x}{\sin^2 x} \).

Step 5: Apply Definition of Cotangent

Recognize that the expression \( -\frac{\cos^2 x}{\sin^2 x} = -\cot^2 x \) as \( \cot x = \frac{\cos x}{\sin x} \).

Key Concepts

Pythagorean identities

Pythagorean identities are fundamental relationships in trigonometry, named after the Pythagorean Theorem. They connect the squares of the basic trigonometric functions: sine, cosine, and tangent. The most famous Pythagorean identity is:

• \( \sin^2 x + \cos^2 x = 1 \)

This identity is essential because it allows you to express any trigonometric function in terms of another. In this exercise, it was used to transform \( \sin^2 x - 1 \) into \( -\cos^2 x \). This is accomplished by subtracting 1 from both sides of the identity:

• \( \sin^2 x - 1 = -\cos^2 x \)

Understanding this relationship helps in simplifying complex trigonometric expressions. By recognizing these identities, we can easily manipulate and reduce expressions to a more workable form. Knowing these identities is like having a key tool in your trigonometry toolkit.

reciprocal identities

Reciprocal identities express trigonometric functions in terms of their reciprocals. These identities are quite handy in simplifying expressions. The primary reciprocal identities are:

• \( \csc x = \frac{1}{\sin x} \)
• \( \sec x = \frac{1}{\cos x} \)
• \( \cot x = \frac{1}{\tan x} \)

In the given exercise, the reciprocal identity \( \csc^2 x = 1 + \cot^2 x \) was used. This identity allowed the transformation of the term \( \cot^2 x + 1 \) into \( \csc^2 x \). These transformations are crucial for expressing complex functions more simply and effectively. By utilizing such identities, one can often simplify expressions dramatically, aiding in easier computation and understanding of trigonometric equations.

trigonometric simplification

Trigonometric simplification is the process of reducing complex trigonometric expressions to simpler forms. This often involves using identities and algebraic manipulation. The main goal is to make the expression easier to understand or compute. In this exercise, several steps were taken to simplify the original expression.The expression \((\sin^2 x - 1)(\cot^2 x + 1)\) was simplified by:

• Using the Pythagorean identity to rewrite \( \sin^2 x - 1 \) as \( -\cos^2 x \).
• Applying the reciprocal identity to express \( \cot^2 x + 1 \) as \( \csc^2 x \).

Then, by using the definition of cosecant and further algebraic manipulation, the expression reduces to:

• \( - \frac{\cos^2 x}{\sin^2 x} \) which equals \( -\cot^2 x \).

These steps reflect the power of simplification in trigonometry, enabling one to transform complicated expressions into simpler equivalents for easier handling and interpretation.

cotangent definition

The definition of cotangent is crucial when dealing with trigonometric simplifications and identities. Cotangent is essentially the reciprocal of the tangent function:

• \( \cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x} \)

This definition was pivotal in the final step of simplifying the given expression. After representing the expression as \(-\frac{\cos^2 x}{\sin^2 x}\), recognizing that this is simply \(-\cot^2 x\) was straightforward.Understanding the definition of cotangent allows one to see how it fits within the wider framework of trigonometric functions. Being aware of such definitions not only helps in simplifying expressions but also in solving trigonometric equations. Grasping these basic yet powerful definitions is essential for efficiently working through and understanding trigonometry-related problems.