Problem 10 - Precalculus: Functions and Graphs

Question

Verify the Identity. $$\frac{1+\csc 3 \beta}{\sec 3 \beta}-\cot 3 \beta=\cos 3 \beta$$

Step-by-Step Solution

Answer

The identity is verified: both sides equal \(\cos 3\beta\).

Step 1: Understanding the Identity

We need to verify the trigonometric identity: \(\frac{1+\csc 3\beta}{\sec 3\beta}-\cot 3\beta=\cos 3\beta\). This means we will show that both sides of the equation are equivalent.

Step 2: Rewrite Reciprocal Trigonometric Functions

Express \(\csc 3\beta\), \(\sec 3\beta\), and \(\cot 3\beta\) in terms of sine, cosine, and tangent. So, \(\csc 3\beta = \frac{1}{\sin 3\beta}\), \(\sec 3\beta = \frac{1}{\cos 3\beta}\), and \(\cot 3\beta = \frac{\cos 3\beta}{\sin 3\beta}\).

Step 3: Substitute and Simplify the Left Side

Substitute these expressions into the left side of the equation. The left side becomes: \(\frac{1 + \frac{1}{\sin 3\beta}}{\frac{1}{\cos 3\beta}} - \frac{\cos 3\beta}{\sin 3\beta}\). This simplifies to \((1 + \csc 3\beta) \cdot \cos 3\beta - \cot 3\beta\).

Step 4: Simplify Further

Multiply \( (1 + \frac{1}{\sin 3\beta}) \cdot \cos 3\beta \) to get \(\cos 3\beta + \frac{\cos 3\beta}{\sin 3\beta}\). Subtract \(\frac{\cos 3\beta}{\sin 3\beta}\), which gives \(\cos 3\beta\).

Step 5: Conclusion

Both sides of the equation are \(\cos 3\beta\), therefore the identity is verified.

Key Concepts

Reciprocal Trigonometric Functions

In trigonometry, reciprocal trigonometric functions are an extension of the fundamental sine, cosine, and tangent functions. The three primary reciprocal functions include cosecant (\( \csc \)), secant (\( \sec \)), and cotangent (\( \cot \)). They are crucial for various calculations and simplifications in trigonometry.

• Cosecant (\( \csc \)) is the reciprocal of sine. Mathematically, \( \csc \theta = \frac{1}{\sin \theta} \).
• Secant (\( \sec \)) is the reciprocal of cosine. So, \( \sec \theta = \frac{1}{\cos \theta} \).
• Cotangent (\( \cot \)) is the reciprocal of tangent, expressed as \( \cot \theta = \frac{1}{\tan \theta} \). Alternatively, it's also \( \frac{\cos \theta}{\sin \theta} \).

Understanding these reciprocals is foundational. They transform complex identities into simpler expressions that are easier to tackle. For example, when you encounter \( \csc 3\beta, \sec 3\beta, \) and \( \cot 3\beta \), rewriting them using these basic reciprocals is often the first step in simplification.

Trigonometric Simplification

Trigonometric simplification is a vital skill in trigonometry, allowing us to bring complex expressions down to simpler forms. By doing this strategically, we turn seemingly difficult problems into more manageable calculations. Start by expressing reciprocal trigonometric functions in terms of sine and cosine.

• Replace \( \csc 3\beta \) with \( \frac{1}{\sin 3\beta} \).
• Convert \( \sec 3\beta \) to \( \frac{1}{\cos 3\beta} \).
• Express \( \cot 3\beta \) as \( \frac{\cos 3\beta}{\sin 3\beta} \).

Once these substitutions are made, performing operations like multiplication and subtraction becomes direct. It's a way of leveraging known identities and basic algebra to clarify pathways to solutions. In our identity, simplifying the left side by submitting these into action yielded an expression that reduced to the expected outcome of \( \cos 3\beta \). The trick often lies in how you arrange and combine these fundamental relationships.

Verifying Identities

Verifying trigonometric identities involves showing that two different-looking expressions are actually the same. It's like solving a puzzle where each piece must fit exactly. The goal is to prove one side of the equation equals the other. Start by simplifying one or both sides using known trigonometric identities and algebraic techniques.

• Identify complex parts and rewrite them using basic trigonometric identities or simplifications.
• Easily navigate through potential algebraic manipulations like factoring or expanding expressions.
• Check each step to ensure logical progression towards making both sides identical.

In the original exercise, this process meant breaking down the left side of the equation into simpler terms. Then, through careful algebra and substitution, aligning it exactly with the right side. Successfully verifying the identity confirms understanding and application of fundamental trigonometric principles. This task not only helps in checking one's own work but also deepens comprehension of the relationships between different trigonometric functions.