Problem 31 - Algebra and Trigonometry

Question

Verify the identity. $$ \sin B+\cos B \cot B=\csc B $$

Step-by-Step Solution

Answer

The identity \( \sin B + \cos B \cot B = \csc B \) is verified.

Step 1: Understand the identity to verify

We need to verify if \( \sin B + \cos B \cot B = \csc B \) is true by simplifying the left-hand side to see if it equals the right-hand side. Our goal is to manipulate the expression on the left to rewrite it in terms of \( \csc B \).

Step 2: Express \( \cot B \) in terms of sine and cosine

The cotangent function is defined as \( \cot B = \frac{\cos B}{\sin B} \). Substitute this into the expression:\[ \sin B + \cos B \cdot \frac{\cos B}{\sin B}. \]

Step 3: Simplify the expression

Distribute \( \cos B \) over \( \cot B \) and simplify:\[\sin B + \frac{\cos^2 B}{\sin B}.\]Combine the terms into a single fraction:\[\frac{\sin^2 B + \cos^2 B}{\sin B}.\]

Step 4: Use the Pythagorean identity

Recall the identity \( \sin^2 B + \cos^2 B = 1 \). Substitute this into the numerator:\[\frac{1}{\sin B}.\]

Step 5: Recognize \( \csc B \)

The definition of the cosecant is \( \csc B = \frac{1}{\sin B} \). Therefore, the expression simplifies to \( \csc B \). This confirms the identity:\[\sin B + \cos B \cot B = \csc B.\]

Step 6: Conclusion

Upon simplification, the left-hand side \( \sin B + \cos B \cot B \) equals the right-hand side \( \csc B \). Therefore, the given identity is verified.

Key Concepts

Understanding the Cosecant Function

The cosecant function, denoted as \( \csc B \), plays a significant role in trigonometry, often appearing in various equations and identities. It is the reciprocal of the sine function, which means that \( \csc B \) is defined mathematically as \( \csc B = \frac{1}{\sin B} \). In practical terms, it is used when dealing with triangles, especially when the sine value is known or easy to find, but its reciprocal is needed in calculations.
Understanding the cosecant function is crucial because it allows you to simplify complex trigonometric expressions. When you see a sine function and you need to convert it into a form that matches another term, like \( \csc \), you recall that simple reciprocal relationship.
A key point to remember is that just like sine, the cosecant function is undefined for values where \( \sin B = 0 \), since division by zero is undefined. This typically happens for angles where \( B = n\pi \), where \( n \) is any integer. Thus, knowing both when to apply the cosecant function and its limitations is essential in trigonometric identities and equations.

Exploring the Cotangent Function

The cotangent function, another essential trigonometric concept, is represented as \( \cot B \) and is defined as the reciprocal of the tangent function. Mathematically, this can be expressed as \( \cot B = \frac{1}{\tan B} = \frac{\cos B}{\sin B} \). This formula is crucial as it allows the conversion of trigonometric expressions into different forms, which is often necessary during verification or simplification tasks.
In many problems, like the one we are discussing, the cotangent function helps break down the equation into sine and cosine components, which are easier to handle and simplify. By substituting \( \cot B = \frac{\cos B}{\sin B} \), you can transform and simplify expressions effectively, allowing you to connect the expression with other identities such as the Pythagorean identity.

• Cotangent is particularly useful in situations where you want to express the angle in terms of functions that are already known or more convenient to work with.
• It is essential to remember that \( \cot B \) is undefined where \( \sin B = 0 \), so care should be taken to ensure that these points are avoided in any computation involving \( \cot B \).

Utilizing the Pythagorean Identity

The Pythagorean identity is a fundamental concept in trigonometry that states \( \sin^2 B + \cos^2 B = 1 \). This identity is derived from the Pythagorean theorem and is valid for any angle \( B \). It is widely used in simplifying and verifying trigonometric expressions, often serving as a key transformation tool.
In problems like the one discussed, the Pythagorean identity provides a shortcut to reduce expressions to known values. For instance, when you encounter an expression like \( \sin^2 B + \cos^2 B \) in the numerator, recognizing the identity immediately simplifies this to \( 1 \). This leap allows you to replace a potentially complex expression with a much simpler form. In the solution provided, this led directly to verifying that the equation reduced to \( \csc B \).

• The Pythagorean identity is useful for manipulating expressions where sine and cosine terms need to be rearranged or simplified.
• Remembering this identity also aids in checking correctness, as any deviation from it often hints at a mistake in calculation or algebraic manipulation.

Mastery of this identity allows for quick simplification of expressions and serves as a cornerstone for many higher-level trigonometric applications.