Problem 10 - Precalculus with Calculus Previews

Question

Use the fundamental identities and the even-odd identities to simplify each expression. $$ 1+\frac{\cot \beta}{\tan \beta} $$

Step-by-Step Solution

Answer

The simplified expression is \( \sec^2 \beta \).

Step 1: Identify Reciprocal Identities

Recall that \( \cot \beta = \frac{1}{\tan \beta} \) by the reciprocal identity. Therefore, rewrite the expression: \[ 1 + \frac{\cot \beta}{\tan \beta} = 1 + \frac{\frac{1}{\tan \beta}}{\tan \beta} \]

Step 2: Simplify the Fraction

Solve the fraction in the expression. Simplify \( \frac{\frac{1}{\tan \beta}}{\tan \beta} \) to get \( \frac{1}{\tan^2 \beta} \). Therefore, the expression becomes:\[ 1 + \frac{1}{\tan^2 \beta} \]

Step 3: Use Pythagorean Identity

Recognize that \( \tan^2 \beta = \frac{\sin^2 \beta}{\cos^2 \beta} \) and that there is a trigonometric identity involving \( \tan^2 \beta \):\[ \sec^2 \beta = 1 + \tan^2 \beta \] Thus, \( \frac{1}{\tan^2 \beta} = \cot^2 \beta \), since \( \cot \beta = \frac{1}{\tan \beta} \). Hence, the expression is:\[ 1 + \cot^2 \beta = \sec^2 \beta \]

Step 4: Confirm Simplified Expression

We have simplified the original expression to \( \sec^2 \beta \), utilizing reciprocal and Pythagorean identities.Therefore, the simplified form of the expression \( 1 + \frac{\cot \beta}{\tan \beta} \) is \( \sec^2 \beta \).

Key Concepts

Reciprocal Identity

In trigonometry, reciprocal identities are an essential tool for simplifying expressions. These identities are based on the reciprocal nature of basic trigonometric functions. A reciprocal simply means flipping the numerator and the denominator of a fraction.

The reciprocal identities you should know are:

• \( \csc \theta = \frac{1}{\sin \theta} \)
• \( \sec \theta = \frac{1}{\cos \theta} \)
• \( \cot \theta = \frac{1}{\tan \theta} \)

In the original exercise, the reciprocal identity \( \cot \beta = \frac{1}{\tan \beta} \) is utilized. This helps in rewriting expressions like \( \frac{\cot \beta}{\tan \beta} \). By expressing cotangent as the reciprocal of tangent, we can replace it with \( \frac{1}{\tan \beta} \). This step is crucial for facilitating further simplification by reducing the complexity of our expressions.

Understanding and applying reciprocal identities can save a significant amount of time and effort when dealing with complex trigonometric expressions.

Pythagorean Identity

The Pythagorean identities are among the fundamental building blocks in trigonometry. They are derived from the Pythagorean theorem, relating the squares of the lengths of the sides of a right triangle. The main identities are:

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

These identities provide powerful tools for simplifying expressions. In the exercise, the identity \( 1 + \tan^2 \beta = \sec^2 \beta \) was used. By recognizing that \( \tan^2 \beta = \frac{\sin^2 \beta}{\cos^2 \beta} \) and substituting back into our expression, the simplification is remarkably straightforward.

The Pythagorean identities are especially beneficial when transforming or equating trigonometric terms within an equation, facilitating the process of achieving simpler equivalent expressions for more manageable calculations.

Trigonometric Simplification

Trigonometric simplification involves the process of transforming a complex expression into a simpler, more manageable form using known identities and algebraic manipulation. This process plays a crucial role in solving trigonometric equations, analyzing angles, and evaluating functions.

In trigonometric simplification, the key is to recognize applicable identities, such as reciprocal and Pythagorean identities, and apply them strategically. Consider the original expression \( 1 + \frac{\cot \beta}{\tan \beta} \). By transforming \( \cot \beta \) to \( \frac{1}{\tan \beta} \) using the reciprocal identity, followed by simplification of the complex fraction, we tend towards simpler terms.

Further incorporation of the Pythagorean identity, \( \tan^2 \beta \), aids in collapsing the expression finally to \( \sec^2 \beta \). This is the essence of trigonometric simplification: continuously applying identities and algebraic transformations until the expression is in its simplest form.

Mastering trigonometric simplification not only makes solving equations more efficient but also improves our overall ability to manipulate trigonometric functions with ease.