Problem 23 - Precalculus Mathematics for Calculus

Question

Simplify the trigonometric expression. $$\frac{2+\tan ^{2} x}{\sec ^{2} x}-1$$

Step-by-Step Solution

Answer

The expression simplifies to \( \cos^2 x \).

Step 1: Recall Trigonometric Identities

We need to recall that \( \tan^2 x + 1 = \sec^2 x \). This identity will be useful in simplifying the expression.

Step 2: Substitute Identity

Substitute the identity \( \tan^2 x + 1 = \sec^2 x \) into the expression \( 2 + \tan^2 x \). So, \( 2 + \tan^2 x = 1 + \sec^2 x + 1 \).

Step 3: Simplify Numerator

Now, substitute back into the expression: \( \frac{2 + \tan^2 x}{\sec^2 x} - 1 \) becomes \( \frac{1 + \sec^2 x + 1}{\sec^2 x} - 1 \).

Step 4: Separate the Fraction

Separate the fraction as: \( \frac{1 + (\sec^2 x + 1)}{\sec^2 x} = \frac{1}{\sec^2 x} + \frac{\sec^2 x + 1}{\sec^2 x} \).

Step 5: Further Simplify the Fractions

Simplify each term: \( \frac{1}{\sec^2 x} = \cos^2 x \) and \( \frac{\sec^2 x + 1}{\sec^2 x} = 1 + \frac{1}{\sec^2 x} = 2 \).

Step 6: Calculate Result

Now subtract 1 from the expression: \( \cos^2 x + 1 - 1 = \cos^2 x \).

Step 7: Final Result

The simplified expression is \( \cos^2 x \). Thus, the trigonometric expression simplifies to \( \cos^2 x \).

Key Concepts

Simplifying Trigonometric Expressions

When we talk about simplifying trigonometric expressions, the goal is to rewrite them in a more manageable or recognizable form. This is typically achieved by using known trigonometric identities or relationships among the trigonometric functions.

• Begin with identifying relevant identities that can transform the original expression. An understanding of trigonometric identities such as the Pythagorean identities can be very useful.
• In the given exercise, the identity \( \tan^2 x + 1 = \sec^2 x \) plays a crucial role. Recognizing such identities helps to break down complex expressions into simpler components.
• The simplification process often involves substitution, often making one part of the expression resemble a trigonometric identity. This can lead to substantial simplification.

Simplifying trigonometric expressions is a valuable skill in solving more complex math problems, and it is especially useful in calculus and physics to make equations easier to solve or interpret.

Trigonometric Functions

Trigonometric functions, including sine, cosine, tangent, and their reciprocals (cosecant, secant, tangent), are fundamental in trigonometry. They relate the angles and sides of a triangle, especially in a right-angled triangle.

• Each function has a unique relationship and can be expressed in terms of others via identities. For example, the secant function, denoted as \( \sec x \), is indeed the reciprocal of the cosine function, \( \cos x \).
• These functions and their identities simplify the complexities of trigonometric expressions and enable the solving of geometric problems. In our exercise, the use of \( \tan^2 x + 1 = \sec^2 x \) stems from these foundational identities.

Understanding trigonometric functions and their interrelations is pivotal for students as it forms the building blocks for advanced math topics.

Algebraic Manipulation in Trigonometry

Algebraic manipulation involves applying algebraic rules to manipulate trigonometric expressions in a step-by-step manner until they are simplified or rewritten in a desired form.

• In trigonometry, algebraic manipulation often involves addition, subtraction, multiplying, dividing, factoring, and expanding terms using trigonometric identities.
• In the given exercise, parts of the expression were separated to further simplify using algebraic manipulation, which includes breaking down complex fractions into simpler parts.
• Simplification can be viewed as a layered process: substituting identities, factoring or multiplying terms, and then reducing until you achieve the simplest form possible.

The ability to seamlessly merge algebra with trigonometry through manipulation techniques is essential for mastering advanced topics in both mathematics and applied sciences.