Problem 82 - Precalculus Mathematics for Calculus

Question

Verify the identity. $$\tan ^{2} x-\cot ^{2} x=\sec ^{2} x-\csc ^{2} x$$

Step-by-Step Solution

Answer

The identity is verified as both sides simplify to \(\frac{\sin^2(x) - \cos^2(x)}{\sin^2(x) \cos^2(x)}\).

Step 1: Express in Terms of Sine and Cosine

Begin by expressing tangent, cotangent, secant, and cosecant in terms of sine \(\sin(x)\) and cosine \(\cos(x)\): - \(\tan(x) = \frac{\sin(x)}{\cos(x)}\) - \(\cot(x) = \frac{\cos(x)}{\sin(x)}\) - \(\sec(x) = \frac{1}{\cos(x)}\) - \(\csc(x) = \frac{1}{\sin(x)}\) Now, square these expressions: - \(\tan^2(x) = \frac{\sin^2(x)}{\cos^2(x)}\) - \(\cot^2(x) = \frac{\cos^2(x)}{\sin^2(x)}\) - \(\sec^2(x) = \frac{1}{\cos^2(x)}\) - \(\csc^2(x) = \frac{1}{\sin^2(x)}\)

Step 2: Write the Identity Using These Representations

Substitute these squared expressions into the identity: \[\frac{\sin^2(x)}{\cos^2(x)} - \frac{\cos^2(x)}{\sin^2(x)} = \frac{1}{\cos^2(x)} - \frac{1}{\sin^2(x)}\]

Step 3: Simplify LHS

To simplify the left-hand side \(\frac{\sin^2(x)}{\cos^2(x)} - \frac{\cos^2(x)}{\sin^2(x)}\), find a common denominator. The common denominator is \(\sin^2(x) \cos^2(x)\). Rewriting each term: \[\frac{\sin^4(x) - \cos^4(x)}{\sin^2(x) \cos^2(x)}\] Using the difference of squares formula, \(a^2 - b^2 = (a-b)(a+b)\): \[\frac{(\sin^2(x) - \cos^2(x))(\sin^2(x) + \cos^2(x))}{\sin^2(x) \cos^2(x)}\] Since \(\sin^2(x) + \cos^2(x) = 1\), this simplifies to: \[\frac{\sin^2(x) - \cos^2(x)}{\sin^2(x) \cos^2(x)}\]

Step 4: Simplify RHS

For the right-hand side \( \frac{1}{\cos^2(x)} - \frac{1}{\sin^2(x)} \), rewrite with a common denominator \(\sin^2(x) \cos^2(x)\) to get: \[\frac{\sin^2(x) - \cos^2(x)}{\sin^2(x) \cos^2(x)}\]

Step 5: Compare Both Sides

Notice that after simplification, both the left-hand side and the right-hand side of the original equation simplify to: \[\frac{\sin^2(x) - \cos^2(x)}{\sin^2(x) \cos^2(x)}\] Since both sides are equal, the given identity is verified.

Key Concepts

Understanding Trigonometric Functions

Trigonometric functions play a crucial role in understanding the relationships between different angles and sides of triangles. They also help describe circular motion. Let's break down some of the basic functions used in solving our exercise:

• Tangent (\( \tan(x)\)): Represents the ratio of the opposite side to the adjacent side in a right triangle. Expressed as \( \frac{\sin(x)}{\cos(x)} \). This can be useful when you need to relate sine and cosine to find angles.
• Cotangent (\( \cot(x)\)): This is the reciprocal of tangent. It is computed as \( \frac{\cos(x)}{\sin(x)} \). It's helpful when you're dealing with ratios in trigonometry.
• Secant (\( \sec(x)\)): Represents the reciprocal of cosine and is defined as \( \frac{1}{\cos(x)} \). It becomes particularly useful when angles grow or shrink drastically.
• Cosecant (\( \csc(x)\)): This is the reciprocal of sine, defined as \( \frac{1}{\sin(x)} \). Cosecant helps when dealing with variations in sine values, especially in different quadrants.

Each of these plays a specific role, and understanding them paves the way for simplifying complex trigonometric expressions.

Simplification Techniques

Simplification is the process of making an expression easier to handle by reducing it to its simplest form. In the context of the given exercise, simplification involves expressing complex trigonometric functions using sine and cosine to reduce complexities. Here are some simplification strategies:

• Common Denominators: When dealing with fractions, find a common denominator, like how we handled \( \frac{\sin^2(x)}{\cos^2(x)} - \frac{\cos^2(x)}{\sin^2(x)} \). This lets us combine the fractions easily.
• Difference of Squares: Used with expressions like \( a^2 - b^2 = (a-b)(a+b) \). This technique breaks down products of squares into simpler components, as seen when simplifying the left-hand side of our equation.
• Fundamental Identities: Knowledge of identities such as \( \sin^2(x) + \cos^2(x) = 1 \) allows for strategic replacements that simplify parts of an expression.

Learning these techniques is invaluable for making trigonometric problems more approachable and solving them more efficiently.

Algebraic Manipulation in Trigonometry

Trigonometry often requires a keen use of algebraic manipulation. This helps in progressively transforming expressions to reach a desired form. Here's how algebraic manipulation was applied in the exercise:

• Substitution: We replaced the trigonometric identities into sine and cosine. This foundational step paved the way for clearer expressions and made further computations simpler.
• Combining and cancelling terms: While simplifying both sides of the identity, we combined terms over a common denominator. This helped to later cancel out parts that appear the same on both sides of the equation, validating the identity.
• Use of Reciprocal Functions: Transformations using the reciprocal identities such as secant and cosecant helped in arranging terms that seem otherwise complicated.

By using algebraic manipulation, the exercise illustrates how to overcome the complexities of trigonometric identities, making the verification of such equations straightforward."}]}]}