Problem 37 - Precalculus: Functions and Graphs

Question

Verify the Identity. $$\frac{1}{\tan \beta+\cot \beta}=\sin \beta \cos \beta $$

Step-by-Step Solution

Answer

The identity is verified as \( \sin \beta \cos \beta \).

Step 1: Express Cotangent in terms of Tangent

First, recall that the cotangent of an angle is the reciprocal of its tangent. Thus, we can write \( \cot \beta = \frac{1}{\tan \beta} \). This will help us simplify the expression.

Step 2: Substitute Cotangent in the Identity

Plug the expression from Step 1 into the left-hand side of the identity. We have \( \frac{1}{\tan \beta + \frac{1}{\tan \beta}} \).

Step 3: Simplify the Denominator

Combine the terms in the denominator to a common fraction: \( \tan \beta + \frac{1}{\tan \beta} = \frac{\tan^2 \beta + 1}{\tan \beta} \).

Step 4: Simplify the Entire Expression

Rewrite the expression using the result from Step 3: \( \frac{1}{\frac{\tan^2 \beta + 1}{\tan \beta}} = \frac{\tan \beta}{\tan^2 \beta + 1} \).

Step 5: Rewrite Using Trigonometric Identity

Recall the trigonometric identity: \( \tan^2 \beta + 1 = \sec^2 \beta \). Substitute: \( \frac{\tan \beta}{\sec^2 \beta} = \tan \beta \cos^2 \beta \).

Step 6: Express Result in Terms of Sine and Cosine

Use \( \tan \beta = \frac{\sin \beta}{\cos \beta} \) to convert \( \tan \beta \cos^2 \beta \) to an equivalent expression in sine and cosine: \( \frac{\sin \beta}{\cos \beta} \times \cos^2 \beta = \sin \beta \cos \beta \).

Step 7: Confirm the Identity

The simplified expression \( \sin \beta \cos \beta \) matches the right-hand side of the identity we wanted to prove. Therefore, \( \frac{1}{\tan \beta + \cot \beta} = \sin \beta \cos \beta \), confirming the identity.

Key Concepts

Tangent and Cotangent

Tangent and cotangent are fundamental trigonometric functions that express relationships in right triangles and periodic oscillations.
The tangent of an angle, denoted as \( \tan \beta \), is defined as the ratio of the opposite side to the adjacent side in a right triangle.
Meanwhile, the cotangent is its reciprocal, so \( \cot \beta = \frac{1}{\tan \beta} \).
These functions are particularly useful because they allow us to express one in terms of the other, as seen in the original exercise.

• Tangent and cotangent help solve complex trigonometric problems by simplifying expressions.
• The reciprocal relationship allows for alternative expressions in verification processes.

Understanding how these functions relate is crucial for solving trigonometric identities, as it provides pathways to simplify and manipulate complex expressions.

Sine and Cosine Relations

Sine and cosine are essential trigonometric functions that relate to the angles and sides of triangles.
They are the building blocks for more complex trigonometric identities, including tangent and cotangent.
Sine, \( \sin \beta \), is the ratio of the opposite side to the hypotenuse, and cosine, \( \cos \beta \), is the ratio of the adjacent side to the hypotenuse.

• Both sine and cosine can be used to express other trigonometric functions.
• The relationship between these two functions is pivotal in altering expressions into more convenient forms.

By expressing \( \tan \beta \) as \( \frac{\sin \beta}{\cos \beta} \), we can leverage this relationship to transform and confirm identities, as showcased in this exercise.
This helps in transitioning complex expressions into simpler sine-cosine equivalents, which are easier to manage.

Simplifying Expressions

Simplifying trigonometric expressions is a key skill in both studying identities and solving equations.
It involves rewriting expressions in a more manageable form, often by utilizing known identities or relationships.

• Breaking down complex expressions to their simpler components is essential for understanding and calculation.
• Efficient simplification leads to quicker verifications and solutions.

In the exercise, simplification is achieved by substituting \( \cot \beta \) and utilizing the identity \( \tan^2 \beta + 1 = \sec^2 \beta \).
Each step aims to reduce the problem into well-known forms, exemplified by expressing the tangent function in terms of sine and cosine.
This process not only confirms the original identity but also strengthens the foundational skills needed for tackling trigonometric problems.