Problem 50 - A Graphical Approach to Precalculus with Limits

Question

Write expression in terms of sine and cosine, and simplify it. (The final expression does not have to be in terms of sine and cosine.) $$(\sec \theta-1)(\sec \theta+1)$$

Step-by-Step Solution

Answer

The expression simplifies to \(\tan^2 \theta\).

Step 1: Recognize the Identity

Recognize that we have the expression \(( ext{sec} \theta - 1)( ext{sec} \theta + 1)\). This is a difference of squares because it matches the form \((a-b)(a+b) = a^2-b^2\) with \(a = \sec \theta\) and \(b = 1\).

Step 2: Apply the Difference of Squares

Apply the difference of squares identity: \(( ext{sec} \theta - 1)( ext{sec} \theta + 1) = \sec^2 \theta - 1^2 = \sec^2 \theta - 1\).

Step 3: Use Trigonometric Identity for Secant

Recall the trigonometric identity \(\sec \theta = \frac{1}{\cos \theta}\). Therefore, \(\sec^2 \theta = \frac{1}{\cos^2 \theta}\). Substitute this into the expression: \(\sec^2 \theta - 1 = \frac{1}{\cos^2 \theta} - 1\).

Step 4: Simplify Using Common Denominator

To simplify \(\frac{1}{\cos^2 \theta} - 1\), express 1 as \(\frac{\cos^2 \theta}{\cos^2 \theta}\) to have a common denominator: \(\frac{1}{\cos^2 \theta} - \frac{\cos^2 \theta}{\cos^2 \theta} = \frac{1 - \cos^2 \theta}{\cos^2 \theta}\).

Step 5: Use Pythagorean Identity

Recall the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\). Therefore, \(1 - \cos^2 \theta = \sin^2 \theta\). Substitute this into the expression to get: \(\frac{\sin^2 \theta}{\cos^2 \theta}\).

Step 6: Simplify Final Expression

The expression \(\frac{\sin^2 \theta}{\cos^2 \theta}\) can be simplified to \((\frac{\sin \theta}{\cos \theta})^2\). Therefore, the expression simplifies to \(\tan^2 \theta\).

Key Concepts

Difference of Squares

The difference of squares is an algebraic identity that is very useful in simplifying expressions. It is given by the formula \((a-b)(a+b) = a^2 - b^2\). This identity works by recognizing that when two binomials with conjugate pairs are multiplied, the middle terms cancel out. In essence, you're left with only the square of the first term minus the square of the second term.
For example, with the expression \((\sec \theta - 1)(\sec \theta + 1)\), we can identify this as a difference of squares. Here, \(a = \sec \theta\) and \(b = 1\).

By applying the identity, we have:
\((\sec \theta)^2 - 1^2 = \sec^2 \theta - 1\).
This simplification is often a first step in solving trigonometric expressions and allows us to subsequently replace the \(\sec^2 \theta\) term using trigonometric identities.
Knowing how to apply the difference of squares is very helpful in algebra and trigonometry.

Secant Function

The secant function, often abbreviated as \(\sec\), is a fundamental trigonometric function. It is the reciprocal of the cosine function, meaning \(\sec \theta = \frac{1}{\cos \theta}\). This function is useful when dealing with trigonometric identities and transformations. In this exercise, the expression \(\sec^2 \theta\) arises.

To further simplify this, we use the reciprocal identity to express \(\sec^2 \theta\) as:
\(\sec^2 \theta = \left(\frac{1}{\cos \theta}\right)^2 = \frac{1}{\cos^2 \theta}\).
Being comfortable with expressions in terms of sine and cosine, especially \(\sec\) in terms of \(\cos\), is pivotal for simplifying more complex expressions and solving trigonometric equations.
This identity helps us incorporate the next crucial identity: the Pythagorean identity, aiding us in simplification.

Pythagorean Identity

The Pythagorean identity is one of the most important identities in trigonometry. It states that \(\sin^2 \theta + \cos^2 \theta = 1\), which helps connect the sine and cosine functions. In the context of this exercise, it's crucial for transforming expressions involving \(\sec^2 \theta\).

If you rearrange the Pythagorean identity, you get:
\(1 - \cos^2 \theta = \sin^2 \theta\).
This transformation is particularly useful when you need to express one trigonometric function in terms of another. For our exercise, substituting \(1 - \cos^2 \theta\) with \(\sin^2 \theta\) leverages this identity, allowing the expression\(\sec^2 \theta - 1\) to be rewritten as \(\sin^2 \theta\) over the common denominator \(\cos^2 \theta\).
This simplification yields the expression:
\(\frac{\sin^2 \theta}{\cos^2 \theta}\), which further simplifies to \(\tan^2 \theta\) since \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).
Mastering the Pythagorean identity and its applications can make working with trigonometric expressions more intuitive and less challenging.