Problem 81 - 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 after substituting \( \sec u = \frac{1}{\cos u} \) and simplifying.

Step 1: Express Secant in Terms of Cosine

Recall that the secant function is the reciprocal of the cosine function. Thus, \( \sec u = \frac{1}{\cos u} \). Substitute this into the left side of the identity: \[ \frac{\sec u - 1}{\sec u + 1} = \frac{\frac{1}{\cos u} - 1}{\frac{1}{\cos u} + 1}. \]

Step 2: Simplify the Expression

Simplify the expression from Step 1: \[ \frac{\frac{1 - \cos u}{\cos u}}{\frac{1 + \cos u}{\cos u}}. \] This expression simplifies to \( \frac{1 - \cos u}{1 + \cos u} \) upon simplifying the complex fraction by multiplying the numerator and the denominator by \( \cos u \).

Step 3: Verify the Simplified Identity

Now that we have \( \frac{1 - \cos u}{1 + \cos u} \) on both sides of the equation, the identity \( \frac{\sec u - 1}{\sec u + 1} = \frac{1 - \cos u}{1 + \cos u} \) is verified.

Key Concepts

Secant Function

The secant function is one of the six primary trigonometric functions, often abbreviated as "sec." It is defined based on the cosine function as its reciprocal. This means that the secant of an angle, denoted as \( \sec u \), is equal to \( \frac{1}{\cos u} \). This relationship highlights the dependency of secant on cosine, making understanding their relationship important.Let's delve into this further:

• Reciprocal Relationship: If cosine \( \cos u \) is known, secant can be easily calculated using \( \sec u = \frac{1}{\cos u} \). If \( \cos u \) is zero, secant is undefined since division by zero is not possible.
• Periodicity: Both secant and cosine functions have periodic behavior. They repeat values over specific intervals of angle \( u \).
• Graphical Insight: On the unit circle, cosine represents the x-coordinate. The secant function maps this value to a length along the line extending from the origin to the point on the cosine wave.

The secant function, like other trigonometric functions, is not valid for all values of its input. Specifically, values where \( \cos u = 0 \) are excluded due to division by zero. Understanding the concept of secant through its link with cosine forms a key part of solving trigonometric identities.

Cosine Function

The cosine function is a cornerstone in trigonometry. It is commonly used to determine the ratio of the adjacent side to the hypotenuse in a right-angled triangle. Represented as \( \cos u \), where \( u \) is the angle in question, its value can be used to find other trigonometric functions through relationships and identities.Here are some fundamental aspects of the cosine function:

• Range: The cosine function can take any angle \( u \) as input and outputs a value between -1 and 1.
• Periodicity: It completes a full cycle every \( 360^{\circ} \) or \( 2\pi \) radians, making it a periodic function.
• Symmetry: Cosine is an even function, meaning \( \cos(-u) = \cos(u) \). This property is useful when simplifying expressions.

In terms of applications, the cosine function is broadly used across fields such as physics, engineering, and geometry. Its periodic nature makes it indispensable in modeling oscillatory systems and wave phenomena. By mastering the cosine function, students gain a deeper understanding of how angles interact with different dimensions and trigonometric identities.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions are critical for simplifying and solving many trigonometric expressions. They "invert" the classic trigonometric functions, leading to new functions such as secant (\( \sec u \)), cosecant (\( \csc u \)), and cotangent (\( \cot u \)).Here's a closer look at these functions:

• Secant: \( \sec u = \frac{1}{\cos u} \) is the reciprocal of cosine. This function is undefined where \( \cos u = 0 \).
• Cosecant: \( \csc u = \frac{1}{\sin u} \) acts as the reciprocal of sine. It is undefined where \( \sin u = 0 \).
• Cotangent: \( \cot u = \frac{1}{\tan u} \) inversely relates to tangent. Where \( \tan u = 0 \), \( \cot u \) cannot exist.

These reciprocal functions reveal new connections in the trigonometric world:

• Application: Reciprocal functions often help simplify complex expressions, proving particularly useful in calculus and advanced trigonometry.
• Completeness: They contribute to a more comprehensive trigonometric toolkit, assisting in solving a broader range of problems.
• Graphical Representation: Graphs of reciprocal functions reflect the nature of their definitions, often mirroring the periods and asymptotes of their base functions.

Understanding and using reciprocal trigonometric functions unlocks the potential to solve numerous mathematical and real-world problems with ease.