Problem 59 - Precalculus: Functions and Graphs

Question

Show that the equation is not an Identity. $$\cos (\sec t)=1$$

Step-by-Step Solution

Answer

The equation is not an identity as it holds true only for certain values of \( t \), not universally.

Step 1: Understand the Components of the Equation

Start by analyzing the given equation: \( \cos(\sec t) = 1 \). Here, \( \sec t \) is the secant of \( t \), which is defined as the reciprocal of cosine: \( \sec t = \frac{1}{\cos t} \). This means \( \cos t \) cannot be zero, as secant would be undefined.

Step 2: Transform \(\sec t\) in Terms of \(\cos t\)

Since \( \sec t = \frac{1}{\cos t} \), substitute \( \sec t \) into the equation: \( \cos\left(\frac{1}{\cos t}\right) = 1 \). This means that \( \frac{1}{\cos t} \), or \( \sec t \), must be an angle where the cosine of that angle equals 1.

Step 3: Analyze When \(\cos x = 1\)

The cosine function equals 1 at specific angles: \( x = 2k\pi \) where \( k \) is an integer. Hence, \( \sec t = 2k\pi \).

Step 4: Check for Validity of the Equation

Substitute back to check if \( \cos(\sec t)=1 \) holds for any \( t \): From Step 3, \( \sec t = 2k\pi \). Cosine of \( 2k\pi \) is 1, meaning only those values of \( t \) make \( \cos(\sec t)=1 \). These values are not true for all \( t \), which indicates the equation is not true universally.

Step 5: Conclusion

A requirement for an identity is that it must hold true for all possible values in its domain. Since \( \cos(\sec t) = 1 \) does not hold for every \( t \) within the domain of \( \cos \) and \( \sec \), it is not an identity.

Key Concepts

Secant Function

The secant function, denoted as \( \sec t \), is an important trigonometric function rooted in the concept of the reciprocal of the cosine function.

• Definition: \( \sec t = \frac{1}{\cos t} \).
• Domain: The secant function is not defined for angles where cosine is zero, such as \( \frac{\pi}{2} + k\pi \) (where \( k \) is an integer) because division by zero is undefined.

The reason it is important to understand the secant function in this context is because we are trying to verify an equation involving both secant and cosine. In simpler terms, whenever you know the cosine of an angle, you can find the secant by taking the reciprocal of that cosine value. Due to its reciprocal nature, the secant function has a range that includes values less than -1 or greater than 1, meaning it stretches beyond the limitations of the cosine function. This distinct behavior makes secant key in evaluating transformations involving reciprocal trigonometric identities.

Cosine Function

Cosine is one of the fundamental trigonometric functions, often used in various identities and transformation equations. Understanding its properties can assist in deciphering more complex trigonometric relationships.

• Definition: \( \cos t \), where \( t \) represents an angle in radians or degrees.
• Range: The function values lie between -1 and 1.
• Primary Angle Property: \( \cos(x) = 1 \) when \( x = 2k\pi \) for any integer \( k \), meaning the angle is a multiple of the full circle (360° or \( 2\pi \) radians).

In this problem, cosine takes a critical role in verifying an equation involving its reciprocal, the secant. It is crucial to understand where the function equals certain values to determine the validity of new identities. In essence, cosine represents the horizontal distance on the unit circle, and knowing when it equals 1 helps us explore the periodic and symmetrical nature of trigonometric functions.

Mathematical Proof

A mathematical proof involves logical reasoning and step-by-step analysis to determine the validity of a statement or equation. In trigonometry, proofs often require a deep understanding of function properties and identities.

• Definition: A systematic argument that verifies the truth or falsehood of a mathematical statement.
• Key Components: Involves assumptions, definitions, quasi-experiments, and logical deductions.
• Identity Verification: Requires a statement to hold true for all permissible values within its domain.

In solving the given problem, we aim to prove or disprove the equation \( \cos(\sec t) = 1 \). The steps involve:- Recognizing the reciprocal nature of secant and cosine.- Expressing secant in terms of cosine and substituting into the equation.- Analyzing specific angle values where \( \cos(x) = 1 \) can occur.However, these specific instances where the equation holds true are not sufficient to confirm it as an identity since identities must be universally valid across the entire domain. Therefore, another essential lesson from this exercise is identifying that mathematical proofs require full coverage of a function's domain to validate as an identity.