Problem 95 - A Graphical Approach to Precalculus with Limits

Question

Verify that equation is an identity. \(\frac{1}{\tan \alpha-\sec \alpha}+\frac{1}{\tan \alpha+\sec \alpha}=-2 \tan \alpha\)

Step-by-Step Solution

Answer

The given equation is an identity because both sides are equal: \( -2 \tan \alpha \).

Step 1: Simplify Left Side

First, rewrite the left-hand side of the equation by finding a common denominator. You have the expression \( \frac{1}{\tan \alpha - \sec \alpha} + \frac{1}{\tan \alpha + \sec \alpha} \). The common denominator is \((\tan \alpha - \sec \alpha)(\tan \alpha + \sec \alpha)\).

Step 2: Apply the Difference of Squares Formula

Simplify the denominator using the difference of squares formula: \[ (\tan \alpha - \sec \alpha)(\tan \alpha + \sec \alpha) = \tan^2 \alpha - \sec^2 \alpha \]Replace \( \sec^2 \alpha \) with \( 1/\cos^2 \alpha \) to further simplify.

Step 3: Simplify the Numerator

The numerators of each fraction are \( \tan \alpha + \sec \alpha \) and \( \tan \alpha - \sec \alpha \), respectively. When combined over the common denominator, you get:\[ (\tan \alpha + \sec \alpha) + (\tan \alpha - \sec \alpha) = 2 \tan \alpha \]

Step 4: Complete Left Side Simplification

Now, the left side becomes: \[ \frac{2 \tan \alpha}{\tan^2 \alpha - \sec^2 \alpha} \]

Step 5: Relate to Trigonometric Identities

Recognize that \( \tan^2 \alpha - \sec^2 \alpha = -(\sec^2 \alpha - \tan^2 \alpha) = -1 \) using the identity \( \sec^2 \alpha = 1 + \tan^2 \alpha \).

Step 6: Final Simplification

So, replace the denominator in the left side equation:\[ \frac{2 \tan \alpha}{-1} = -2 \tan \alpha \] which matches the right side of the given equation.

Key Concepts

Trigonometric Functions

Trigonometric functions such as sine, cosine, tangent, and secant form the foundation for understanding trigonometric identities.

• Sine (\(\sin\)) and Cosine (\(\cos\)): These are the fundamental trigonometric functions representing the ratio of sides in a right triangle corresponding to an angle.
• Tangent (\(\tan\)): It is defined as the ratio of the sine to the cosine of an angle, \(\tan \alpha = \frac{\sin \alpha}{\cos \alpha}\).
• Secant (\(\sec\)): It is the reciprocal of the cosine function, given by \(\sec \alpha = \frac{1}{\cos \alpha}\).

These functions are interrelated and can be converted into each other using various identities. Understanding these relationships is vital in manipulating and verifying identities like the one in the exercise.

Algebraic Manipulation

Algebraic manipulation involves combining and rearranging terms to simplify an equation or expression. In our exercise, this skill is used to simplify the left side of the equation.Firstly, a common denominator for the expressions is found. This step involves creating a single expression from two fractions, which require the \((\tan \alpha - \sec \alpha)(\tan \alpha + \sec \alpha)\) as a common base.
Second, the denominator is simplified using the difference of squares formula, which states \((a-b)(a+b) = a^2 - b^2\). In this exercise, it becomes \(\tan^2 \alpha - \sec^2 \alpha\).Lastly, recognizing that the numerator simplifies to \(2 \tan \alpha\), the equation can be neatly managed into a more straightforward expression.

• These steps illustrate the power of algebraic manipulation in breaking down complex expressions into understandable components.
• It shows how equations simplify when trigonometric identities are cleverly applied and manipulated.

Verification of Identities

Verification of identities is the process of proving that one expression is equal to another, using known identities and operations in mathematics. In this exercise, we began with simplifying one side of an equation and showing how it equals the other.

• Use of Identities: The exercise employs the Pythagorean identity \(\sec^2 \alpha = 1 + \tan^2 \alpha\), which is pivotal in transforming the left side.
• Direct Comparison: The goal is to manipulate the equation until both sides appear identical.
• Final Adjustments: By acknowledging that \(\tan^2 \alpha - \sec^2 \alpha = -1\), it's possible to simplify the fraction further, reaching the identity desired.

Verification reflects a structured approach to establish relationships between different mathematical components, harnessing both trigonometric knowledge and algebraic skills. It confirms validity through rigorous logical steps, culminating in a satisfying equivalence that demonstrates the truth of the expression.