Problem 35 - Precalculus: Functions and Graphs

Question

Verify the Identity. $$\frac{\tan \alpha}{1+\sec \alpha}+\frac{1+\sec \alpha}{\tan \alpha}=2 \csc \alpha$$

Step-by-Step Solution

Answer

The identity is verified.

Step 1: Rewrite Terms on Left Side

Start with the given expression \( \frac{\tan \alpha}{1+\sec \alpha} + \frac{1+\sec \alpha}{\tan \alpha} \) and aim to simplify it. Rewrite \( \tan \alpha = \frac{\sin \alpha}{\cos \alpha} \) and \( \sec \alpha = \frac{1}{\cos \alpha} \) to make substitutions later.

Step 2: Substitute Terms

Substitute \( \tan \alpha = \frac{\sin \alpha}{\cos \alpha} \) and \( \sec \alpha = \frac{1}{\cos \alpha} \) back into the expression: \[ \frac{\frac{\sin \alpha}{\cos \alpha}}{1+\frac{1}{\cos \alpha}} + \frac{1+\frac{1}{\cos \alpha}}{\frac{\sin \alpha}{\cos \alpha}} \].

Step 3: Simplify Each Fraction

Simplify \( \frac{\frac{\sin \alpha}{\cos \alpha}}{1+\frac{1}{\cos \alpha}} \) as follows: multiply the numerator and denominator by \( \cos \alpha \) to get \( \frac{\sin \alpha}{\cos \alpha + 1} \). Similarly, simplify \( \frac{1+\frac{1}{\cos \alpha}}{\frac{\sin \alpha}{\cos \alpha}} \) to \( \frac{\cos \alpha + 1}{\sin \alpha} \).

Step 4: Add the Simplified Fractions

Find a common denominator to add \( \frac{\sin \alpha}{\cos \alpha + 1} + \frac{\cos \alpha + 1}{\sin \alpha} \). The common denominator is \( \sin \alpha (\cos \alpha + 1) \). Express each term with this denominator: \[ \frac{\sin^2 \alpha + (\cos \alpha + 1)^2}{\sin \alpha (\cos \alpha + 1)} \].

Step 5: Simplify the Numerator

Expand and combine the terms of the numerator \( \sin^2 \alpha + (\cos \alpha + 1)^2 \): \( \sin^2 \alpha + \cos^2 \alpha + 2\cos \alpha + 1 \). Utilize the Pythagorean identity \( \sin^2 \alpha + \cos^2 \alpha = 1 \) to simplify it to \( 2 + 2\cos \alpha \).

Step 6: Factor and Simplify Further

Factor the numerator as \( 2(1 + \cos \alpha) \): \[ \frac{2(1 + \cos \alpha)}{\sin \alpha (\cos \alpha + 1)} \]. Simplify the expression by canceling \( 1 + \cos \alpha \) in the numerator and denominator: \( \frac{2}{\sin \alpha} \).

Step 7: Express as Cosecant

Recognize that \( \frac{2}{\sin \alpha} = 2 \csc \alpha \), confirming that the original expression was correctly transformed into the right side of the identity: \( 2 \csc \alpha \). Both sides are equal.

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a cornerstone concept in trigonometry. It states that for any angle \( \alpha \), the sum of the square of the sine and cosine functions equals one: \( \sin^2 \alpha + \cos^2 \alpha = 1 \). This identity is essential for simplifying trigonometric expressions and solving equations. It directly relates to our exercise and is used in simplifying the numerator in step 5 of our solution.

This identity allows us to express one trigonometric function in terms of another. For instance, if you know the sine of an angle, you can determine its cosine without resorting to measurements. This property is particularly useful in simplifying complex expressions where multiple trigonometric terms are involved. By substituting \( \sin^2 \alpha \) with \( 1 - \cos^2 \alpha \), or vice versa, you can reduce terms and make equations easier to handle.

Cosecant Function

The cosecant function, or \( \csc \alpha \), is the reciprocal of the sine function. It is defined as \( \csc \alpha = \frac{1}{\sin \alpha} \). Understanding this relationship is vital as it allows us to transform expressions involving sine into ones involving cosecant.

In our exercise, recognizing that \( \frac{2}{\sin \alpha} \) can be rewritten as \( 2 \csc \alpha \) was crucial in confirming the identity. This transformation used the definition of the cosecant function, demonstrating its utility in simplifying and verifying trigonometric identities.

The cosecant function often appears in solution processes where dividing by sine is beneficial, as it can simplify expressions and provide clearer insight into the relationships between different trigonometric functions.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions involves breaking down complex expressions into more manageable terms. This can often mean changing formats or using identities and reciprocal relationships to make the expression clearer or simpler.

In the exercise, several key techniques were employed:

• Substituting expressions using known identities for \( \tan \alpha = \frac{\sin \alpha}{\cos \alpha} \) and \( \sec \alpha = \frac{1}{\cos \alpha} \).
• Finding a common denominator for addition, which unified the terms into a single expression.
• Simplifying the complex fraction by canceling out common factors in the numerator and denominator.
• Leveraging the Pythagorean identity to replace \( \sin^2 \alpha + \cos^2 \alpha \) with 1, reducing the equation.
• Utilizing the cosecant function to transform the expression into the desired form \( 2 \csc \alpha \).

These steps highlight the elegance and flexibility of trigonometric identities in transforming expressions, making them approachable and straightforward.