Problem 19 - Algebra 2

Question

Simplify each trigonometric expression. $$ \sec ^{2} \theta-\tan ^{2} \theta $$

Step-by-Step Solution

Answer

The simplified form of the given trigonometric expression \(\sec ^{2} \theta - \tan ^{2} \theta\) is \(1\).

Step 1: Recall the Pythagorean Identity

Firstly, recall the Pythagorean trigonometric identity that relates the secant and tangent functions. This identity is \( \sec ^{2} \theta = \tan^{2} \theta + 1\).

Step 2: Substitute the Identity

Now, substitute this identity into the original expression. Given that \( \sec ^{2} \theta = \tan^{2} \theta + 1\), the expression \( \sec ^{2} \theta - \tan^{2} \theta\) substitutes to \( \tan^{2} \theta + 1 - \tan^{2} \theta\).

Step 3: Simplify the Expression

The tangent terms cancel out in the revised expression, leaving just the constant. Hence, \( \tan^{2} \theta + 1 - \tan^{2} \theta = 1\).

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental equations involving trigonometric functions that are true for every value in their domain. These identities serve as vital tools in simplifying expressions and solving trigonometric equations. They help relate the six trigonometric functions to one another.
Among these identities, you often find

• Pythagorean identities
• Angle sum and difference identities
• Double angle identities
• Reciprocal identities

Using these identities, one can transform complex trigonometric expressions into simpler forms. This simplification is key in solving mathematical problems more efficiently. As we simplify expressions or solve equations, identifying the suitable identity to apply is an essential skill for students to develop.

Pythagorean Identity

The Pythagorean Identity is one of the cornerstone relationships in trigonometry. It comes from the Pythagorean theorem applied in a unit circle context. A commonly used form is: \[\sec ^{2} \theta = \tan^{2} \theta + 1\] This identity derives from the more familiar: \[\sin^{2} \theta + \cos^{2} \theta = 1\] The Pythagorean identities link trigonometric functions through these equalities and allow for convenient substitutions.
By rearranging and substituting one identity into an expression, more complex expressions can be drastically simplified. For example, by replacing \(\sec ^{2} \theta\) with \(\tan^{2} \theta + 1\) in any trigonometric expression, we can reduce it to much simpler forms – revealing hidden relationships between functions.

Secant and Tangent Functions

Secant and tangent are trigonometric functions often used in combination with identities for simplification. The secant function, denoted as \(\sec \theta\), is the reciprocal of the cosine function: \[\sec \theta = \frac{1}{\cos \theta}\]The tangent function, \(\tan \theta\), represents the ratio of sine to cosine:\[\tan \theta = \frac{\sin \theta}{\cos \theta}\]These two functions, when squared, are interconnected through the Pythagorean identity mentioned earlier.
In the simplification process from our exercise, recognizing that

• \(\sec^{2} \theta = \tan^{2} \theta + 1\)

enables the reduction of the expression to a constant "1" by effectively canceling out terms. Understanding how these functions interact and applying identities to simplify expressions is crucial in mastering trigonometry.