Problem 55 - A Graphical Approach to Precalculus with Limits

Question

Verify that each equation is an identity. $$\frac{\cos (A-B)}{\cos A \sin B}=\tan A+\cot B$$

Step-by-Step Solution

Answer

The equation is an identity.

Step 1: Understand the Identity

Before verifying the identity, recall the identity you need to prove: \( \frac{\cos(A-B)}{\cos A \sin B} = \tan A + \cot B \). Also understand the trigonometric identities you might use, such as \( \tan A = \frac{\sin A}{\cos A} \) and \( \cot B = \frac{\cos B}{\sin B} \).

Step 2: Use Cosine Difference Identity

The cosine difference identity is \( \cos(A-B) = \cos A \cos B + \sin A \sin B \). Substitute this into the left-hand side of the equation: \( \frac{\cos A \cos B + \sin A \sin B}{\cos A \sin B} \).

Step 3: Simplify the Left-Hand Side

Separate the terms in the numerator: \( \frac{\cos A \cos B}{\cos A \sin B} + \frac{\sin A \sin B}{\cos A \sin B} \). This can be further simplified into two fractions: \( \frac{\cos B}{\sin B} + \frac{\sin A}{\cos A} \).

Step 4: Simplify Each Fraction

Recognize that \( \frac{\cos B}{\sin B} = \cot B \) and \( \frac{\sin A}{\cos A} = \tan A \). Therefore, the expression becomes \( \cot B + \tan A \).

Step 5: Verify Identity

Now compare the result \( \tan A + \cot B \) obtained from simplifying the left-hand side with the right-hand side. Since they are identical, the expression is verified as an identity.

Key Concepts

Cosine Difference Identity

The cosine difference identity is a useful trigonometric formula that helps us simplify expressions involving the cosine of the difference between two angles. When we have two angles, say \( A \) and \( B \), their cosine difference is given by the identity:

• \( \cos(A - B) = \cos A \cos B + \sin A \sin B \)

This identity allows us to express the cosine of a difference as a sum of products of cosine and sine functions of the individual angles.
It plays a crucial role in transforming trigonometric expressions, verifying identities, and solving trigonometric equations in exercises like the one you are working on.

• Understand the structure: The cosine difference identity depicts adding two products to get the cosine of a difference.
• Application: Substitute \( \cos(A - B) \) using the identity when simplifying expressions.

By substituting \( \cos(A - B) = \cos A \cos B + \sin A \sin B \), you can often break down complex expressions into simpler components, making it easier to verify identities or solve equations.

Tangent Function

The tangent function, denoted as \( \tan \), is fundamental in trigonometry. It describes the ratio of the sine to the cosine for a given angle. For an angle \( A \), the tangent function is defined as:

• \( \tan A = \frac{\sin A}{\cos A} \)

The tangent function is particularly useful because of how it links together the sine and cosine functions in a single ratio.
This makes the tangent function essential for solving trigonometric equations where one might need to relate angles and sides in right-angled triangles.

• Remember the Ratio: The tangent of an angle is the sine divided by the cosine.
• Utility: \( \tan A \) is used in finding slopes in geometry and modeling periodic phenomena in physics.

When verifying identities, recognizing expressions like \( \frac{\sin A}{\cos A} \) can help you identify places where the tangent function can simplify calculations or verifications.

Cotangent Function

The cotangent function, represented as \( \cot \), is also a crucial trigonometric tool, closely related to the tangent. For an angle \( B \), the cotangent function is the reciprocal of the tangent function:

• \( \cot B = \frac{\cos B}{\sin B} \)

This function provides the ratio of the cosine to the sine, offering a different perspective compared to the tangent function.
The cotangent function comes in handy, especially when dealing with complementary angles or simplifying expressions involving ratios of cosine to sine.

• Reciprocal Nature: \( \cot \) is the inverse of \( \tan \), which means \( \cot B = \frac{1}{\tan B} \).
• Solving Identities: Like the tangent function, identifying \( \frac{\cos B}{\sin B} \) as \( \cot B \) can simplify expressions and aid in verifying identities.

Understanding the use of the cotangent function in tandem with other trigonometric identities can be key to efficiently solving trigonometric equations and proving identities.