Problem 47 - 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. \( \dfrac{\tan \theta \cot \theta}{\sec \theta} \)

Step-by-Step Solution

Answer

\( \cos \theta \)

Step 1: Apply Fundamental Identities

Begin with the given expression \( \dfrac{\tan \theta \cot \theta}{\sec \theta} \) and apply the fundamental identities to simplify. The identities to apply are: \\[ \tan \theta = \frac{\sin \theta}{\cos \theta}, \cot \theta = \frac{\cos \theta}{\sin \theta}, \sec \theta = \frac{1}{\cos \theta} \\] After applying these identities, the expression becomes: \\[ \dfrac{\left( \frac{\sin \theta}{\cos \theta} \right) \left( \frac{\cos \theta}{\sin \theta} \right)}{ \frac{1}{\cos \theta}} \\]

Step 2: Reduce the Expression

Now, simplify the expression. Notice that the expressions of \( \sin \theta \) and \( \cos \theta \) cancel in the numerator, which leaves us with: \\[ \dfrac{1}{ \frac{1}{\cos \theta}} \\]

Step 3: Solve the Expression

The fraction in the denominator can be simplified by 'inverting the division', thus resulting in: \\[ \cos \theta \\]

Key Concepts

Fundamental Identities

Fundamental identities are the building blocks of trigonometry, providing the shorthand versions of various equations to ease calculations and problem-solving. When dealing with trigonometric expressions, these identities allow you to rewrite expressions in different forms, making them simpler to manage. Some of the most commonly used fundamental trigonometric identities include:

• Reciprocal identities, such as \( an heta = rac{ ext{opposite}}{ ext{adjacent}} \) and \( an heta = rac{ ext{sin} heta}{ ext{cos} heta} \).
• Quotient identities, where \( an heta = rac{ ext{sin} heta}{ ext{cos} heta} \) is expressed when using sine and cosine functions.
• Pythagorean identities, like \( ext{sin}^2 heta + ext{cos}^2 heta = 1 \).

Each of these identities can be applied to transform and simplify complex trigonometric expressions. Practice using these identities regularly to become proficient at recognizing and applying them in different contexts.

Trigonometric Simplification

Trigonometric simplification is a process of breaking down complex trigonometric expressions into simpler or more manageable forms, often utilizing fundamental identities. This technique is valuable in finding solutions more efficiently.In simplification, several strategies are typically applied:

• Substitute equivalent identities to reduce fractions, like \( an heta = rac{ ext{sin} heta}{ ext{cos} heta} \).
• Cancel out common terms in numerators and denominators to simplify ratios.
• Rewriting complex fractions by simplifying individual components.

By simplifying expressions, you can make equations easier to solve or better understand the relationships between different trigonometric functions. Regular practice and familiarity help in identifying simpler forms quickly.

Trigonometric Functions

Trigonometric functions form the foundation of trigonometry and are used to describe relationships in right-angle triangles and oscillatory processes like waves. These functions relate angles of a triangle to the ratios of its sides.The six main trigonometric functions are:

• Sine (\( ext{sin} \)), cos(\( heta \)): representing the ratio of the opposite side over hypotenuse.
• Cosine (\( ext{cos} \)), cos(\( heta \)): ratio of the adjacent side over hypotenuse.
• Tangent (\( ext{tan} \)), tan(\( heta \)): the ratio of opposite over adjacent sides.

Alongside their reciprocal and co-functions (\( ext{sec} \), \( ext{cosec} \), and \( ext{cot} \)), these trigonometric functions help solve problems related to angles and dimensions. They are crucial in fields like physics, engineering, and computer graphics.

Reciprocal Identities

Reciprocal identities are crucial in trigonometry, as they relate three sets of trigonometric functions: sine, cosine, and tangent, with their respective reciprocals (cosecant, secant, and cotangent).These identities are particularly useful when simplifying trigonometric expressions:

• The secant (\( ext{sec} \)) is the reciprocal counterpart to cosine (\( ext{cos} \)), expressed as \( ext{sec} heta = rac{1}{ ext{cos} heta} \).
• The cosecant (\( ext{csc} \)) is the reciprocal of sine (\( ext{sin} \)), giving \( ext{csc} heta = rac{1}{ ext{sin} heta} \).
• The cotangent (\( ext{cot} \)) is the reciprocal of tangent (\( ext{tan} \)), or \( ext{cot} heta = rac{1}{ ext{tan} heta} \).

These identities simplify expressions by converting reciprocals into terms that are often easier to interpret or further reduce. Understanding and remembering these identities help to solve complex trigonometric equations more efficiently.