Problem 83 - Algebra and Trigonometry

Question

Verify the identity. $$ \frac{\sec u-1}{\sec u+1}=\frac{1-\cos u}{1+\cos u} $$

Step-by-Step Solution

Answer

The identity is verified as both sides equal \( \frac{1 - \cos u}{1 + \cos u} \).

Step 1: Rewrite Secant in Terms of Cosine

Recognize that \( \sec u = \frac{1}{\cos u} \). Substitute this expression into the given identity to start verification: \( \frac{\frac{1}{\cos u} - 1}{\frac{1}{\cos u} + 1} \).

Step 2: Simplify the Numerator and Denominator

Simplify each part separately by finding a common denominator which is \( \cos u \). The expression becomes \( \frac{\frac{1 - \cos u}{\cos u}}{\frac{1 + \cos u}{\cos u}} \).

Step 3: Simplify the Overall Expression

When you divide by a fraction, multiply by its reciprocal: \( \frac{1 - \cos u}{1 + \cos u} \times \frac{\cos u}{\cos u} \). Simplifying gives \( \frac{1 - \cos u}{1 + \cos u} \), which matches the right side of the identity.

Key Concepts

Secant Function

The secant function is an essential part of trigonometry, often represented as \( \sec u \). It is defined as the reciprocal of the cosine function. In formulaic terms, \( \sec u = \frac{1}{\cos u} \).
This means that for any angle \( u \), if you know the value of the cosine, you can quickly find the secant by taking its reciprocal.

• Graphically, the secant function takes on values that are either greater than 1 or less than -1, as cosine values range between -1 and 1.
• The secant function can become very large when the cosine approaches zero, leading to vertical asymptotes in its graph where the function is undefined.
• The secant function, like other trigonometric functions, is periodic with a period of \( 2\pi \), meaning it repeats its pattern every \( 2\pi \) units.

Understanding the role of the secant function, especially in terms of its relationship with the cosine, is crucial for verifying trigonometric identities.

Cosine Function

The cosine function, symbolized by \( \cos u \), represents one of the basic trigonometric functions. It gives the horizontal coordinate of a point on the unit circle corresponding to an angle \( u \).
Here are some key points about the cosine function:

• The cosine function oscillates between -1 and 1. It is symmetrical about the vertical axis, making it an even function, formally expressed as \( \cos(-u) = \cos u \).
• It is periodic with a period of \( 2\pi \), meaning the behavior of cosine values repeats itself every \( 2\pi \) radians.
• The cosine function is pivotal when dealing with the Pythagorean identity, \( \, \cos^2 u + \sin^2 u = 1 \, \). This identity frequently serves as a foundation for verifying more complex trigonometric identities.

Knowing the behavior and properties of the cosine function helps you understand its reciprocal, the secant function, and lets you manipulate trigonometric expressions effectively.

Verify Identities

Verifying trigonometric identities involves proving that two expressions are equivalent for all angles where the functions are defined. It is a critical process in understanding and simplifying trigonometric equations.
To verify the identity \( \frac{\sec u-1}{\sec u+1}=\frac{1-\cos u}{1+\cos u} \), we rely on expressing functions in terms of their fundamental forms, like rewriting secant as \( \sec u = \frac{1}{\cos u} \).

• Start by substituting known identities to express everything in terms of sine and cosine, the most basic trigonometric functions.
• Look for opportunities to factorize or simplify fractions by using common trigonometric techniques, such as multiplying through by conjugates or finding common denominators.
• Check each step carefully to ensure transformations preserve equivalence between the sides of the equation.

Mastering the verification of identities enhances your comprehension of trigonometric relationships and provides a foundation for solving more complex problems.