Problem 114 - A Graphical Approach to Precalculus with Limits

Question

Verify that each equation is an identity by using any of the identities introduced in the first three sections of this chapter. $$\frac{1}{\sec t-1}+\frac{1}{\sec t+1}=2 \cot t \csc t$$

Step-by-Step Solution

Answer

The equation is an identity.

Step 1: Substitute Reciprocal Identities

The secant function can be expressed in terms of cosine: \( \sec t = \frac{1}{\cos t} \). Substitute \( \sec t \) with \( \frac{1}{\cos t} \) in the original equation: \[ \frac{1}{\frac{1}{\cos t} - 1} + \frac{1}{\frac{1}{\cos t} + 1} = 2 \cot t \csc t. \]

Step 2: Simplify the Denominators

Rewrite the denominators: \( \frac{1}{\cos t} - 1 = \frac{1 - \cos t}{\cos t} \) and \( \frac{1}{\cos t} + 1 = \frac{1 + \cos t}{\cos t} \). Substitute these into the equation: \[ \frac{\cos t}{1 - \cos t} + \frac{\cos t}{1 + \cos t}. \]

Step 3: Find a Common Denominator

The common denominator of \( \frac{\cos t}{1 - \cos t} \) and \( \frac{\cos t}{1 + \cos t} \) is \((1-\cos t)(1+\cos t) = 1 - \cos^2 t = \sin^2 t \) (difference of squares). Rewriting gives: \[ \frac{\cos t (1+\cos t) + \cos t (1-\cos t)}{\sin^2 t}. \]

Step 4: Simplify the Numerator

Distribute \( \cos t \) in the numerator: \( \cos t (1 + \cos t) = \cos t + \cos^2 t \) and \( \cos t (1 - \cos t) = \cos t - \cos^2 t \). Combine terms: \( \cos t + \cos^2 t + \cos t - \cos^2 t = 2\cos t \).

Step 5: Rewrite the Expression

Substitute back: \[ \frac{2 \cos t}{\sin^2 t} = 2 \frac{\cos t}{\sin^2 t}. \] Recognize the identity: \( \cot t = \frac{\cos t}{\sin t} \) and \( \csc t = \frac{1}{\sin t} \). Thus, \( \frac{\cos t}{\sin^2 t} = \cot t \csc t \).

Step 6: Confirm the Identity

Rewrite \( 2 \frac{\cos t}{\sin^2 t} \) as \( 2 \cot t \csc t \). This confirms \[ \frac{1}{\sec t - 1} + \frac{1}{\sec t + 1} = 2 \cot t \csc t \].

Key Concepts

Reciprocal Identities

In trigonometry, reciprocal identities are essential building blocks that allow us to express one function in terms of another. The reciprocal identity involves expressing secant, cosecant, and cotangent in terms of the more primary functions: sine, cosine, and tangent. For example:

• Secant is the reciprocal of cosine: \( \sec t = \frac{1}{\cos t} \)
• Cosecant is the reciprocal of sine: \( \csc t = \frac{1}{\sin t} \)
• Cotangent is the reciprocal of tangent: \( \cot t = \frac{1}{\tan t} \)

These identities are powerful tools for transforming and simplifying trigonometric expressions. By substituting these reciprocal expressions into the original functions, we can begin to unravel complicated trigonometric identities and pave the way for further algebraic manipulation. Recognizing when and how to use these identities can greatly aid in solving and proving equations.

Trigonometric Functions

Trigonometric functions are among the basic functions in mathematics, essential for understanding the relationships within a right triangle. They are periodic, meaning they repeat their values in regular intervals, which is especially useful in applications involving cycles or waves.
The primary trigonometric functions are sine, cosine, and tangent.

• Sine of an angle is the ratio of the opposite side to the hypotenuse: \( \sin t \)
• Cosine is the ratio of the adjacent side to the hypotenuse: \( \cos t \)
• Tangent is the ratio of the opposite side to the adjacent side: \( \tan t \)

From these arise the other three functions, which are considered derived or reciprocal functions: secant, cosecant, and cotangent. Understanding these foundational functions allows us to explore a wide range of mathematical problems, including the verification of trigonometric identities.

Proof of Identities

Proving trigonometric identities involves validating that two different expressions are equivalent for all values in their domains. This proof is important because it confirms relationships between the trigonometric functions themselves, allowing us to derive more complex equations.
The step-by-step process generally includes:

• Substituting identities (e.g., reciprocal identities) to simplify expressions.
• Finding common denominators if fractions are involved.
• Simplifying expressions using basic algebra and known identities, such as the Pythagorean identity \( \sin^2 t + \cos^2 t = 1 \).

During the proof, manipulate one or both sides of the equation until they are identical, thus confirming they represent the same quantity. This process emphasizes the importance of knowing foundational identities and using them to breakdown and reconstruct expressions.

Algebraic Manipulation

Algebraic manipulation in trigonometry involves rearranging and simplifying expressions using known identities and algebraic operations like distributing, factoring, and combining like terms. This manipulation is crucial for simplifying expressions and solving equations.
Consider the example from the problem: Substituting the secant identity transformed the expression into a more workable form using cosine, allowing the expression to be simplified. Once the denominators were found and simplified by recognizing it as a difference of squares, it was easier to manage.
Algebraic simplification often includes:

• Rewriting expressions using identities, such as \( \sec t = \frac{1}{\cos t} \).
• Expanding or distributing terms, e.g., multiplying out \( \cos t(1 + \cos t) \).
• Combining and simplifying terms to recognize like terms and simplify expressions into useful identities, such as \( 2 \cot t \csc t \).

Mastering these techniques is vital in mathematics, enabling learners to confidently tackle more challenging trigonometric equations and proofs.