Problem 17 - Algebra and Trigonometry

Question

Simplify the trigonometric expression. $$ \frac{\sec ^{2} x-1}{\sec ^{2} x} $$

Step-by-Step Solution

Answer

The expression simplifies to \( \sin^2 x \).

Step 1: Analyze the expression

We start by looking at the given expression \( \frac{\sec^{2} x - 1}{\sec^{2} x} \). We notice that the expression involves \( \sec^{2} x \). We know from trigonometric identities that \( \sec^{2} x = 1 + \tan^{2} x \). Substitute this identity into the expression where possible.

Step 2: Substitute the identity

Using the identity \( \sec^{2} x = 1 + \tan^{2} x \), we substitute \( \sec^{2} x \) in the expression: \[\frac{(1 + \tan^{2} x) - 1}{1 + \tan^{2} x}.\]Simplify the numerator to get \( \tan^{2} x \): \[\frac{\tan^{2} x}{1 + \tan^{2} x}.\]

Step 3: Recognize the structure

In this step, we recognize that the expression \( \frac{\tan^{2} x}{1 + \tan^{2} x} \) can be related to simpler trigonometric functions. Suppose we rewrite this expression using the identity \( 1 + \tan^{2} x = \sec^{2} x \) again. Notice that \( \frac{\tan^2 x}{\sec^2 x} \) simplifies to \( \sin^2 x \).

Step 4: Final simplification

At this point, notice that \( \frac{\tan^{2} x}{1 + \tan^{2} x} \) implies dividing \( \tan^{2} x \) by \( \sec^2 x \). Knowing \( \tan x = \frac{\sin x}{\cos x} \) and \( \sec x = \frac{1}{\cos x} \), we have:\[\frac{\left(\frac{\sin x}{\cos x}\right)^2}{\left(\frac{1}{\cos x}\right)^2} = \sin^2 x.\]

Step 5: Conclusion: Solution found

Thus, the simplified form of the given expression is \( \sin^2 x \).

Key Concepts

Trigonometric Simplification

Trigonometric simplification involves rewriting complex trigonometric expressions into simpler forms. The goal is to make the expression easier to understand or compute. It often requires the use of trigonometric identities.The expression \( \frac{\sec^{2} x - 1}{\sec^{2} x} \) can be simplified by recognizing patterns and substituting known identities. For example, substituting \( \sec^{2} x = 1 + \tan^{2} x \) helps transform the expression. Simplification often involves basic algebraic manipulations like combining like terms or reducing fractions.When simplifying, remember:

• Identify and use appropriate trigonometric identities.
• Simplify the numerator and denominator separately before combining them.
• Look for factorable expressions that can be reduced.

Keeping these steps in mind can significantly simplify complex problems in trigonometry.

Secant Function

The secant function, denoted as \( \sec x \), is one of the fundamental trigonometric functions. It is defined as the reciprocal of the cosine function: \( \sec x = \frac{1}{\cos x} \).In the realm of trigonometric identities, \( \sec^2 x \) often appears. One crucial identity involving the secant function is \( \sec^2 x = 1 + \tan^2 x \). This identity is immensely helpful in transforming and simplifying expressions where secant and tangent functions appear together.Remember:

• \( \sec x \) remains undefined when \( \cos x = 0 \), due to division by zero.
• \( \sec x \) is periodic with a period of \( 2\pi \).
• It is an even function, meaning \( \sec(-x) = \sec x \).

Understanding the properties of the secant function is crucial for enhancing one's ability to tackle trigonometric problems.

Tangent Function

The tangent function, noted as \( \tan x \), is another key trigonometric function. It is defined as the ratio of the sine to the cosine function, \( \tan x = \frac{\sin x}{\cos x} \).This function is frequently encountered in trigonometric identities and equations. The identity \( 1 + \tan^2 x = \sec^2 x \) is particularly notable. This identity allows conversion between expressions involving tangent and secant, aiding in simplification.Key traits of \( \tan x \) include:

• It has vertical asymptotes where \( \cos x = 0 \), which means \( \tan x \) is undefined.
• \( \tan x \) is periodic with a period of \( \pi \).
• It is an odd function, satisfying \( \tan(-x) = -\tan x \).

These characteristics make the tangent function an essential part of solving trigonometric expressions.