Problem 67 - Algebra 2

Question

Show that \((\sec \theta+1)(\sec \theta-1)=\tan ^{2} \theta\) is an identity.

Step-by-Step Solution

Answer

By manipulating the left hand side of the given equation, using the reciprocal relationship of secant and cosine, and the Pythagorean identity, we have proven that indeed \((\sec \theta+1)(\sec \theta-1)=\tan ^{2} \theta \), showing this expression is an identity.

Step 1:Multiply the left-hand side (LHS) of the equation

By using the property of difference of two squares for \((\sec \theta+1)(\sec \theta-1)\) , the expression simplifies to \( (\sec ^{2} \theta - 1) \).

Step 2: Substitute the secant function

We can substitute \(\sec ^{2} \theta\) with \(1/\cos ^{2} \theta \). Thus, the expression becomes \((1/\cos ^{2} \theta - 1)\).

Step 3: Rewrite the expression

Rewrite the expression with common denominator. It becomes \( \frac{1- \cos ^{2} \theta}{\cos ^{2} \theta} \).

Step 4: Using the Pythagorean identity

We use the Pythagorean identity \(\sin ^{2} \theta + \cos ^{2} \theta = 1\) to replace \(1- \cos ^{2} \theta\) with \(\sin ^{2} \theta\). Thus, the expression becomes \( \frac{\sin ^{2} \theta}{\cos ^{2} \theta} \).

Step 5: Simplify the expression

The ratio of sine to cosine is tangent: \(\tan \theta = \sin \theta / \cos \theta\). Thus, the expression becomes \( \tan ^{2} \theta\) which is equal to the right hand side (RHS).

Key Concepts

Difference of Squares

The difference of squares is a useful algebraic identity to know, especially for simplifying expressions like \((a + b)(a - b)\). This identity states that \((a + b)(a - b) = a^2 - b^2\).
Here's why it's helpful:

• It allows complex terms to be simplified quickly.
• It's the foundation for many algebraic manipulations.

In this exercise, we apply the difference of squares to \((\sec \theta + 1)(\sec \theta - 1)\), simplifying it to \(\sec^2 \theta - 1\).
This step is crucial in connecting trigonometric identities, especially once we incorporate the Pythagorean Identity.

Pythagorean Identity

The Pythagorean Identity is one of the fundamental trigonometric identities. It states:\(\sin^2 \theta + \cos^2 \theta = 1\).
This relation is crucial for transforming and simplifying trigonometric expressions.

• By rearranging the identity, \(\cos^2 \theta = 1 - \sin^2 \theta\), we can substitute into other expressions.
• In this problem, the identity helps us express \(1 - \cos^2 \theta\) as \(\sin^2 \theta\), which further simplifies our expression.

Understanding this identity makes it easier to grasp other trigonometric relationships and solve complex problems.It’s like a magic key that unlocks a lot of trigonometric puzzles!

Secant Function

The secant function, denoted as \(\sec \theta\), is one of the reciprocal trigonometric functions.Here's what you need to know:

• It’s the reciprocal of the cosine function: \(\sec \theta = \frac{1}{\cos \theta}\).
• It helps express trigonometric functions in terms of sine and cosine, which are the more basic trigonometric functions.

In this exercise, we substitute \(\sec^2 \theta\) with \(\frac{1}{\cos^2 \theta}\) to prepare the equation for the application of the Pythagorean Identity.
Recognizing the secant function's role in these equations is critical to solving trigonometric identities effectively.

Tangent Function

The tangent function is another critical trigonometric function, represented by \(\tan \theta\).Here’s how it works:

• It represents the ratio of sine to cosine: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).
• Its square, \(\tan^2 \theta\), plays an essential role when simplifying expressions—like in this exercise.

At the end of the problem, the transformation \(\frac{\sin^2 \theta}{\cos^2 \theta}\) to \(\tan^2 \theta\) completes our demonstration. The goal was to show that the simplified version of the left side of the equation actually equaled \(\tan^2 \theta\), confirming that \((\sec \theta+1)(\sec \theta-1)=\tan ^{2} \theta\) is indeed a valid trigonometric identity.