Problem 46 - Precalculus with Limits

Question

In Exercises 37 - 58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer. \( \dfrac{1}{\tan^2 x + 1} \)

Step-by-Step Solution

Answer

Therefore, the simplified form of the expression \( \dfrac{1}{\tan^2 x + 1} \) is \( \cos^2 x \).

Step 1: Identifying fundamental identity

Firstly, identify the Pythagorean identity that can be used to simplify the expression. In this case, \( \tan^2x + 1 = \sec^2x \).

Step 2: Application of fundamental identity

Substitute the identified identity in place of \( \tan^2x + 1 \) in the given expression. Therefore, the initial expression \( \dfrac{1}{\tan^2 x + 1} \) becomes \( \dfrac{1}{\sec^2 x} \).

Step 3: Simplification of expression

Finally, simplify the expression. Since \( \sec x \) is \( \dfrac{1}{\cos x} \), \( \sec^2 x \) will be \( \dfrac{1}{\cos^2 x} \). Thus, \( \dfrac{1}{\sec^2 x} \) becomes \( \cos^2 x \).

Key Concepts

Pythagorean Identity in Trigonometry

The Pythagorean identity is a fundamental concept in trigonometry used to simplify complex expressions. It's one of the cornerstones alongside sine and cosine calculations. The identity comes in three primary forms, which are derivatives of the original:

• \( \sin^2 x + \cos^2 x = 1 \)
• \( 1 + \tan^2 x = \sec^2 x \)
• \( 1 + \cot^2 x = \csc^2 x \)

These identities allow us to transform and simplify trigonometric expressions by substituting equivalent terms. For example, in this exercise, knowing that \( \tan^2 x + 1 = \sec^2 x \) helps us reduce the complexity of the original expression. With these transformations, it becomes easier to work with and further simplify the problem at hand.

The Art of Trigonometric Simplification

Trigonometric simplification is about transforming convoluted trigonometric expressions into easier, more manageable forms. This process involves using trigonometric identities, like Pythagorean identities, to replace complex terms with their simpler equivalents.When simplifying \( \dfrac{1}{\tan^2 x + 1} \), we utilized the identity \( \tan^2 x + 1 = \sec^2 x \). By substituting \( \sec^2 x \) in place of \( \tan^2 x + 1 \), the expression reduces to \( \dfrac{1}{\sec^2 x} \).To simplify further, consider that \( \sec x = \dfrac{1}{\cos x} \). From this relation, \( \sec^2 x = \dfrac{1}{\cos^2 x} \) is derived, which makes \( \dfrac{1}{\sec^2 x} = \cos^2 x \). This transformation process is driven by the need to express the result in standard trigonometric terms, making it easier to integrate into further calculations and applications.

Understanding the Secant Function

The secant function, denoted \( \sec x \), is one of the six primary trigonometric functions and offers unique properties useful in various mathematical applications. The secant function is defined as the reciprocal of the cosine function:\[ \sec x = \dfrac{1}{\cos x} \]Its primary purpose is to simplify expressions where dividing by cosine would be cumbersome due to division rules.In trigonometric identities, \( \sec x \) often appears alongside the tangent function, as seen in the Pythagorean identity \( 1 + \tan^2 x = \sec^2 x \). Understanding \( \sec x \) in this context is crucial for rewriting expressions and solving problems that involve angles and their trigonometric relationships.The secant function helps to bridge various aspects of trigonometry with geometry, calculus, and other advanced mathematical disciplines, enabling a seamless flow of calculations and interpretations.