Problem 79 - Algebra and Trigonometry

Question

Verify the identity. $$ \frac{1}{\sec x+\tan x}+\frac{1}{\sec x-\tan x}=2 \sec x $$

Step-by-Step Solution

Answer

The given identity is verified: \( \frac{1}{\sec x+\tan x}+\frac{1}{\sec x-\tan x}=2 \sec x \).

Step 1: Rewrite Secant and Tangent

We start by rewriting the secant and tangent functions in terms of sine and cosine: \( \sec x = \frac{1}{\cos x} \) and \( \tan x = \frac{\sin x}{\cos x} \). Substitute these into the expression.

Step 2: Combine Fractions

Combine the terms \( \sec x + \tan x \) and \( \sec x - \tan x \) using the common denominator, which is \( \cos x \). This simplifies the fractions to \( \frac{1}{\frac{1+\sin x}{\cos x}} + \frac{1}{\frac{1-\sin x}{\cos x}} \).

Step 3: Simplify Each Fraction

Invert the fractions from Step 2, resulting in \( \frac{\cos x}{1+\sin x} + \frac{\cos x}{1-\sin x} \).

Step 4: Common Denominator

Find a common denominator for the two fractions: \( (1+\sin x)(1-\sin x) = 1 - \sin^2 x = \cos^2 x \). Rewrite each fraction over this common denominator: \( \frac{\cos x(1-\sin x)}{\cos^2 x} + \frac{\cos x(1+\sin x)}{\cos^2 x} \).

Step 5: Add the Numerators

Add the numerators: \( \cos x(1-\sin x) + \cos x(1+\sin x) = \cos x - \cos x \sin x + \cos x + \cos x \sin x = 2 \cos x \).

Step 6: Simplify the Expression

The result from Step 5 is \( \frac{2 \cos x}{\cos^2 x} \). Simplify this to \( \frac{2}{\cos x} \), which is \( 2 \sec x \), thereby verifying the identity.

Key Concepts

Secant Function

The secant function, denoted as \( \sec x \), is a crucial trigonometric identity that represents the reciprocal of the cosine function. Simply put, if you know the value of the cosine, you can find the secant by calculating its reciprocal:

• \( \sec x = \frac{1}{\cos x} \)

The secant function is particularly useful in trigonometric simplifications and verifications as it aids in converting expressions involving angles into a form that is easier to manipulate. It often appears in problems where you need to move from a fraction involving cosine to a simpler expression.

Tangent Function

The tangent function, denoted as \( \tan x \), is another fundamental trigonometric identity frequently encountered in algebra and calculus. This function gives you the ratio between the sine and cosine of an angle:

• \( \tan x = \frac{\sin x}{\cos x} \)

Understanding the tangent is essential when exploring relationships between different trigonometric functions since it frequently appears alongside secant in identities and equations. In verification exercises, expressing tangent in terms of sine and cosine makes it easier to find common denominators or simplify complex expressions.

Trigonometric Simplification

Trigonometric simplification involves rewriting complex trigonometric expressions in a simpler or more useful form. In many exercises, this requires recognizing common patterns or identities, such as what we've done by converting secant and tangent into expressions of sine and cosine. Simplification strategies include:

• Using reciprocal identities, like \( \sec x = \frac{1}{\cos x} \)
• Recognizing and employing Pythagorean identities, such as \( 1 - \sin^2 x = \cos^2 x \)

Knowing how to simplify an expression is key in trigonometric calculus and algebra, as it allows easier manipulation and understanding of functions.

Sine and Cosine

Sine and cosine are the foundational building blocks for all other trigonometric functions. The sine function, \( \sin x \), represents the vertical coordinate on the unit circle, while cosine, \( \cos x \), represents the horizontal. Their relationship is pivotal:

• \( \sin^2 x + \cos^2 x = 1 \)

This identity, known as the Pythagorean identity, allows you to transform and simplify expressions through squaring and manipulation. Understanding the sine and cosine functions is crucial for any trigonometric exploration, helping to bridge relationships among trigonometric expressions.

Verifying Identities

Verifying identities is the process of proving that two trigonometric expressions are equivalent for all values of the variable within the domain. It is a fundamental skill in trigonometry.

• Convert all functions to sine and cosine, if possible.
• Find a common denominator to combine fractions or expressions.
• Simplify systematically, step by step, using known identities.

By using these techniques, verification becomes a structured approach to problem-solving, allowing you to demonstrate the equivalence or validity of a proposed equation.