Problem 29 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-50: Verify the identity. $$ \frac{1+\csc \beta}{\cot \beta+\cos \beta}=\sec \beta $$

Step-by-Step Solution

Answer

The identity is verified: the equation simplifies to \( \sec \beta \).

Step 1: Express Everything in Terms of Sine and Cosine

First, express the trigonometric functions in terms of sine and cosine. Recall that \( \csc \beta = \frac{1}{\sin \beta} \), \( \cot \beta = \frac{\cos \beta}{\sin \beta} \), and \( \sec \beta = \frac{1}{\cos \beta} \). Substitute these into the given identity:\[\frac{1 + \frac{1}{\sin \beta}}{\frac{\cos \beta}{\sin \beta} + \cos \beta} \]

Step 2: Simplify the Numerator

Simplify the numerator, \( 1 + \frac{1}{\sin \beta} \). Combine the terms over a common denominator:\[ 1 + \frac{1}{\sin \beta} = \frac{\sin \beta}{\sin \beta} + \frac{1}{\sin \beta} = \frac{\sin \beta + 1}{\sin \beta} \]

Step 3: Simplify the Denominator

Simplify the denominator, \( \frac{\cos \beta}{\sin \beta} + \cos \beta \). Combine the terms over a common denominator:\[\frac{\cos \beta}{\sin \beta} + \cos \beta = \frac{\cos \beta}{\sin \beta} + \frac{\cos \beta \cdot \sin \beta}{\sin \beta} = \frac{\cos \beta + \cos \beta \cdot \sin \beta}{\sin \beta} \]

Step 4: Substitute Back and Simplify

Substitute the simplified expressions back into the original equation:\[\frac{\frac{\sin \beta + 1}{\sin \beta}}{\frac{\cos \beta + \cos \beta \cdot \sin \beta}{\sin \beta}} = \frac{\sin \beta + 1}{\cos \beta} \]Here, the \( \sin \beta \) terms cancel out. Simplify the result:\[ \frac{\sin \beta + 1}{\cos \beta} = \frac{1}{\cos \beta} = \sec \beta \]This shows that the original equation simplifies to \( \sec \beta \).

Step 5: Verify the Equality

We have shown that both sides simplify to the same expression \( \sec \beta \). Therefore, the original identity is verified to be true.

Key Concepts

Sine and Cosine Functions

Sine and cosine are the fundamental building blocks of trigonometry, these functions allow us to relate angles to side lengths in right triangles. In the context of our problem, we express other trigonometric functions in terms of sine and cosine. Here’s why:

• Sine, \( \sin \beta \): Gives the ratio of the length of the opposite side to the hypotenuse in a right triangle.
• Cosine, \( \cos \beta \): Offers the ratio of the adjacent side to the hypotenuse.

When expressing functions like cosecant, cotangent, and secant, we rely on these main functions:\[ \csc \beta = \frac{1}{\sin \beta}, \quad \cot \beta = \frac{\cos \beta}{\sin \beta}, \quad \sec \beta = \frac{1}{\cos \beta} \]Replacing every trigonometric function with sine and cosine helps in simplifying complex expressions and solving identities, which is a technique widely used to prove the given trigonometric identity verifications.

Trigonometric Simplification

Trigonometric simplification involves breaking down complex expressions into simpler forms. With our exercise, once we express everything in sine and cosine, we focus on simplifying both the numerator and the denominator.
For the numerator:

• Initially look something like: \( 1 + \frac{1}{\sin \beta} \).
• Convert into a single fraction by finding a common denominator: \( \frac{\sin \beta + 1}{\sin \beta} \).

The denominator follows a similar procedure:

• Start off as: \( \frac{\cos \beta}{\sin \beta} + \cos \beta \).
• Unifies to a single fraction: \( \frac{\cos \beta + \cos \beta \cdot \sin \beta}{\sin \beta} \).

Once combined, further simplifications such as cancelling out terms can lead us to verifying or disproving the identity in question, as demonstrated with simplification revealing both sides equate to the secant function.

Cosecant and Secant

Among the notable trigonometric functions are cosecant and secant, which derive from sine and cosine, respectively:

• Cosecant \( \beta \): It is the reciprocal of sine, given by \( \csc \beta = \frac{1}{\sin \beta} \). This function is particularly useful when dealing with fractions in trigonometric identities.
• Secant \( \beta \): Known as the reciprocal of cosine, \( \sec \beta = \frac{1}{\cos \beta} \). It simplifies expressions in which cosine appears in the denominator.

In our exercise, converting to cosecant and secant allows us to simplify and ultimately verify the trigonometric identity. Each step of simplification uses these relationships:

• Understanding reciprocity: Knowing that secant and cosecant are reciprocals of cosine and sine, respectively, helps transform and manipulate expressions efficiently.
• Verifying identities: By showing both sides equal \( \sec \beta \), the original equation holds true.

Recognizing the interplay between these functions is key to unlocking the full potential of trigonometric simplifications and verifications.