Problem 10 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-50: 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: the expression simplifies to \( \cos 3\beta \).

Step 1: Rewrite the Terms

We start with the expression \( \frac{1+\csc 3\beta}{\sec 3\beta}-\cot 3\beta \). Rewrite \( \csc 3\beta \) as \( \frac{1}{\sin 3\beta} \), \( \sec 3\beta \) as \( \frac{1}{\cos 3\beta} \), and \( \cot 3\beta \) as \( \frac{\cos 3\beta}{\sin 3\beta} \). This gives: \[ \frac{1 + \frac{1}{\sin 3\beta}}{\frac{1}{\cos 3\beta}} - \frac{\cos 3\beta}{\sin 3\beta}.\]

Step 2: Simplify the Fraction

Combine the terms in the numerator of the first fraction: \(1 + \frac{1}{\sin 3\beta} = \frac{\sin 3\beta + 1}{\sin 3\beta}\). Substitute back to simplify the first fraction: \[\frac{\sin 3\beta + 1}{\sin 3\beta}\times \cos 3\beta = \cos 3\beta\cdot \frac{\sin 3\beta + 1}{\sin 3\beta}.\]

Step 3: Simplify the Expression Further

Distribute \( \cos 3\beta \) across the numerator: \[ \cos 3\beta\cdot \frac{\sin 3\beta + 1}{\sin 3\beta} = \frac{\cos 3\beta \cdot \sin 3\beta + \cos 3\beta}{\sin 3\beta}. \] This simplifies to: \[ \frac{\cos 3\beta \sin 3\beta}{\sin 3\beta} + \frac{\cos 3\beta}{\sin 3\beta}. \]

Step 4: Subtract the Remaining Term

Rewrite \(\cot 3\beta \) as \( \frac{\cos 3\beta}{\sin 3\beta} \) and subtract from the expression: \[ \frac{\cos 3\beta \sin 3\beta}{\sin 3\beta} + \frac{\cos 3\beta}{\sin 3\beta} - \frac{\cos 3\beta}{\sin 3\beta}. \] The two identical terms cancel out, leaving: \( \cos 3\beta. \)

Step 5: Verify the Identity

Our final expression is \( \cos 3\beta \), which is exactly what we needed to verify in the identity \( \frac{1+\csc 3 \beta}{\sec 3 \beta}-\cot 3 \beta = \cos 3 \beta \). Hence, the identity is verified.

Key Concepts

Cosecant

The cosecant function, often abbreviated as \( \csc \theta \), is the reciprocal of the sine function. This means if \( \sin \theta = y \), then \( \csc \theta = \frac{1}{y} \). It is used commonly in trigonometric identities and simplification of expressions where the sine function is involved. Being the reciprocal, it shares the same domain restrictions as sine, except that \( \theta \) should not be angles where \( \sin \theta = 0 \), as division by zero is undefined. In the step-by-step solution, you will notice the use of \( \csc 3\beta \) being rewritten as \( \frac{1}{\sin 3\beta} \). This is a fundamental approach to simplify expressions involving trigonometric terms, as recognizing these reciprocals aids in further algebraic manipulation.

Secant

The secant function, denoted as \( \sec \theta \), is the reciprocal of the cosine function, i.e., \( \sec \theta = \frac{1}{\cos \theta} \). This relationship allows you to transform expressions involving secant into ones that involve only sine and cosine, making it easier to simplify complex trigonometric identities. In our exercise, \( \sec 3\beta \) is rewritten as \( \frac{1}{\cos 3\beta} \) to facilitate the multiplication and simplification steps.Understanding and recognizing trigonometric functions as reciprocals is a key step in verifying equations and identities. This fundamental understanding helps in algebraic manipulation required to bring expressions to their simplest form.

Cotangent

The cotangent function, expressed as \( \cot \theta \), is the reciprocal of the tangent function. It can be defined as \( \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \).In many problems, including this exercise, rewriting cotangent in terms of sine and cosine aids the simplification process. For instance, the transformation \( \cot 3\beta = \frac{\cos 3\beta}{\sin 3\beta} \) allowed for direct subtraction of terms in the simplified expression.By rewriting trigonometric functions, complex expressions become more manageable, and unnecessary terms often cancel out, simplifying the process of verifying trig identities.

Cosine

The cosine function, denoted by \( \cos \theta \), is a fundamental trigonometric function. It represents the x-coordinate of a point on the unit circle at an angle \( \theta \) from the positive x-axis.In trigonometric identities, cosine often partners with sine in expressions. It is crucial in many trig identities, such as the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \). In the given exercise, our goal was to simplify the original expression ultimately to \( \cos 3\beta \). Recognizing the target function can guide your simplification steps and verify correctness throughout the process.

Simplification of Expressions

Simplification of trigonometric expressions is an essential skill that requires replacing complex terms with simpler equivalents using known identities.Here are some techniques to simplify expressions:

• Rewriting trigonometric functions in terms of sine and cosine.
• Using reciprocal identities, like converting \( \sec \theta \) to \( \frac{1}{\cos \theta} \).
• Combining fractions with common denominators where possible.
• Cancelling out matching terms after transformation.

In our step-by-step solution, each operation – from rewriting to combining and cancelling terms – served the purpose of simplifying the expression to match the initial identity goal: \( \cos 3\beta \). Understanding these steps allows for efficient verification of identities and solutions of trigonometric problems.