Problem 51 - Precalculus with Limits

Question

In Exercises 37 - 58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer. \( \cos\left(\dfrac{\pi}{2} - x\right) \sec x \)

Step-by-Step Solution

Answer

The simplified expression is \( \tan x \)

Step 1: Switch to sine function

Recognize that using co-function identity, \( \cos\left(\dfrac{\pi}{2} - x\right) = \sin x \). So, the given expression becomes \( \sin x \sec x\)

Step 2: Substitute secant function

Switch from secant to its reciprocal function. Hence, \( \sec x = \dfrac{1}{\cos x} \). Thus, the expression transforms to \( \sin x \cdot \dfrac{1}{\cos x} \)

Step 3: Simplify the Expression

Simplify the product to achieve the final expression. So, \( \sin x \cdot \dfrac{1}{\cos x} = \tan x \)

Key Concepts

Fundamental Identities

The fundamental identities in trigonometry are essential tools that provide a basis for simplifying trigonometric expressions.
These identities express trigonometric functions in terms of each other, allowing for the transformation of complex expressions into simpler forms. Among the most crucial are the Pythagorean identities, quotient identities, and reciprocal identities.
For example, the Pythagorean identity states that for any angle x, the square of the sine function plus the square of the cosine function equals one, mathematically expressed as \( \text{sin}^2 x + \text{cos}^2 x = 1 \).
Understanding and applying these identities can significantly ease solving trigonometry problems, making the seemingly daunting task of simplification much more manageable.

Co-function Identity

The co-function identity relates to functions of complementary angles. In trigonometry, complementary angles sum up to \( \frac{\pi}{2} \) or 90 degrees.
A co-function identity pairs sine with cosine, tangent with cotangent, and secant with cosecant. For instance, \( \sin(\frac{\pi}{2} - x) \) is equal to \( \cos(x) \), and vice versa.

Examples of Co-function Identity

• \( \cos(\frac{\pi}{2} - x) = \sin(x) \)
• \( \tan(\frac{\pi}{2} - x) = \cot(x) \)
• \( \sec(\frac{\pi}{2} - x) = \csc(x) \)

Employing co-function identities allows one to transform expressions and simplify calculations in trigonometry, thereby solving problems more efficiently.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions are merely the inverse of the primary trigonometric functions. These are secant (\( \sec \)), cosecant (\( \csc \)), and cotangent (\( \cot \)) which are the reciprocals of cosine, sine, and tangent, respectively.

• \( \sec x = \frac{1}{\cos x} \)
• \( \csc x = \frac{1}{\sin x} \)
• \( \cot x = \frac{1}{\tan x} \)

Understanding these relationships is key to simplifying trigonometric expressions. For instance, if you encounter a secant function in a problem, knowing that it can be written as the reciprocal of the cosine function can lead to further simplifications.

Trigonometric Simplification

Simplifying trigonometric expressions is a common task in mathematics, which involves reducing expressions to their simplest form using trigonometric identities and properties.
The process typically includes recognizing patterns, substituting equivalent expressions, and performing algebraic manipulations.

Common Simplification Steps

• Identify and apply fundamental identities
• Use co-function identities to express functions of complementary angles
• Transform expressions using reciprocal functions
• Cancel common factors
• Combine and reduce fractions whenever possible

Through these steps, expressions like \( \cos(\frac{\pi}{2} - x) \sec x \) can be simplified to \( \tan x \) by recognizing that cosine of an angle's complement is the sine of that angle and by substituting the reciprocal of cosine for secant.