Problem 29 - Precalculus: Functions and Graphs

Question

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

Step-by-Step Solution

Answer

The identity is verified: \( \frac{1+\csc \beta}{\cot \beta+\cos \beta} = \sec \beta \).

Step 1: Express Trigonometric Functions in Terms of Sine and Cosine

To verify the identity, we will rewrite each trigonometric function in terms of sine and cosine. We have:- \( \csc \beta = \frac{1}{\sin \beta} \)- \( \cot \beta = \frac{\cos \beta}{\sin \beta} \)- \( \sec \beta = \frac{1}{\cos \beta} \).

Step 2: Simplify the Numerator

Substitute \( \csc \beta \) with \( \frac{1}{\sin \beta} \) in the numerator:\[ 1 + \csc \beta = 1 + \frac{1}{\sin \beta} = \frac{\sin \beta + 1}{\sin \beta} \].

Step 3: Simplify the Denominator

Substitute \( \cot \beta \) with \( \frac{\cos \beta}{\sin \beta} \) and leave \( \cos \beta \) as is. Combine under a common denominator:\[ \cot \beta + \cos \beta = \frac{\cos \beta}{\sin \beta} + \cos \beta = \frac{\cos \beta + \cos \beta \sin \beta}{\sin \beta} = \frac{\cos \beta (1 + \sin \beta)}{\sin \beta} \].

Step 4: Simplify the Fraction

Now, substitute the simplified numerator and denominator into the fraction on the left side of the identity:\[ \frac{1 + \csc \beta}{\cot \beta + \cos \beta} = \frac{\frac{\sin \beta + 1}{\sin \beta}}{\frac{\cos \beta (1 + \sin \beta)}{\sin \beta}} = \frac{\sin \beta + 1}{\sin \beta} \times \frac{\sin \beta}{\cos \beta (1 + \sin \beta)} \].

Step 5: Cancel Common Terms

Cancel \( \sin \beta + 1 \) from the numerator and the denominator:\[ \frac{\sin \beta + 1}{\sin \beta + 1} = 1 \]. Thus, we are left with:\[ \frac{1}{\cos \beta} \].

Step 6: Verify the Result

The simplified expression \( \frac{1}{\cos \beta} \) matches the right-hand side of the identity \( \sec \beta \). Therefore, the given identity is verified to be true.

Key Concepts

Sine and Cosine Functions

Let's dive into the sine and cosine, two essential trigonometric functions often used in identities and equations. These functions arise from the relationships within a right triangle. The sine of an angle is the ratio of the length of the opposite side to the hypotenuse, whereas the cosine is the ratio of the adjacent side to the hypotenuse.
When dealing with trigonometric functions, it is useful to express everything in terms of sine and cosine. This can often reveal connections and simplify complex expressions.

• Sine ( \( ext{sin} \beta\)): Useful in determining heights and depths, given the hypotenuse and angle. Expressed as a basic ratio.
• Cosine (\( ext{cos} \beta\)): Helps find the horizontal distance using the hypotenuse and angle. Also a fundamental ratio.

Utilizing the sine and cosine functions for converting expressions allows for straightforward manipulation and simplification, leading to even verifying complex trigonometric identities.

Trigonometric Simplification

Trigonometric simplification is the process of reducing trigonometric expressions to simpler forms or proving identities. Simplification often involves rewriting expressions using known identities or transforming them into terms of just sine and cosine, which allows for better manipulation.
This can include processes like substituting reciprocal identities. For example, \(\csc \beta = \frac{1}{\sin \beta}\), \(\cot \beta = \frac{\cos \beta}{\sin \beta}\), and \(\sec \beta = \frac{1}{\cos \beta}\). Such identities help in transforming the expression into a more recognizable form, easing the process of simplification or identity verification.

• Combine like terms or expressions under a common denominator.
• Simplify fractions by canceling common factors, much like simplifying algebraic expressions.
• Use Pythagorean identities like \( \sin^2 \beta + \cos^2 \beta = 1 \) if needed for additional simplification.

Reducing expressions to simplest forms is both a powerful and essential tool in solving trigonometric problems, helping in both learning and practical applications alike.

Verifying Identities

Verifying trigonometric identities involves showing that an equation holds true for all possible angle values, converting one side of the equation to equal the other using known identities and properties. It's like proving that two different routes lead to the same destination.
In this verification process, breaking down each part into recognizable terms helps immensely. Common strategies include expressing all terms in sine and cosine, simplifying numerators and denominators, and carefully reducing complex fractions. With these steps, you can effectively verify complex identities.
To verify an identity:

• Start by expressing all trigonometric functions in terms of sine and cosine.
• Simplify the more complicated side using identities, common denominators, and factoring.
• Check if the simplified expression matches the other side of the equation.

The exercise of verifying identities not only develops critical math skills but also deepens understanding of trigonometric functions and their interrelationships. This methodical process is foundational in mastering trigonometry.