Problem 22 - Algebra 2 and Trigonometry

Question

In \(3-26,\) prove that each equation is an identity. $$ \cos \theta+\frac{\sin ^{2} \theta}{1+\cos \theta}=1 $$

Step-by-Step Solution

Answer

The equation \( \cos \theta + \frac{\sin^2 \theta}{1 + \cos \theta} = 1 \) is an identity.

Step 1: Understand the Identity

The goal is to show that the equation \( \cos \theta + \frac{\sin^2 \theta}{1+\cos \theta} = 1 \) holds true for all values of \( \theta \). This means the left side and right side of the equation must be equal under all valid conditions.

Step 2: Rewrite \( \sin^2 \theta \)

Recall the Pythagorean identity: \( \sin^2 \theta = 1 - \cos^2 \theta \). Substitute this into the expression: \( \cos \theta + \frac{1 - \cos^2 \theta}{1 + \cos \theta} \).

Step 3: Simplify the Fraction

The expression inside the fraction is \( \frac{1 - \cos^2 \theta}{1 + \cos \theta} \). Factor the numerator: \( 1 - \cos^2 \theta = (1 - \cos \theta)(1 + \cos \theta) \). Substitute this into the fraction: \( \frac{(1 - \cos \theta)(1 + \cos \theta)}{1 + \cos \theta} \).

Step 4: Cancel Terms

Cancel the \( 1 + \cos \theta \) in the numerator and denominator: \( \frac{1 - \cos \theta}{1} = 1 - \cos \theta \).

Step 5: Simplify the Expression

Combine this result with \( \cos \theta \): \( \cos \theta + (1 - \cos \theta) = 1 \). This simplifies the left side of the equation to match the right side, which is \( 1 \).

Step 6: Conclusion

Since the left side simplifies to equal the right side \(1\), we have shown that the given equation is indeed an identity.

Key Concepts

Pythagorean Identity

The Pythagorean identity is one of the cornerstones of trigonometry. It states that for any angle \( \theta \), the sum of the squares of the sine and cosine of that angle is always equal to one. Mathematically, it is expressed as:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)

This identity is extremely useful when simplifying expressions that involve sine and cosine. In the original exercise, we used the Pythagorean identity to rewrite \( \sin^2 \theta \) as \( 1 - \cos^2 \theta \). This substitution simplifies the process of proving the equation is an identity. It's a powerful tool because it enables us to convert between the sine and cosine functions using this equivalence. By understanding and applying the Pythagorean identity, any expression involving \( \sin^2 \theta \) can be transformed into an expression of cosine, and vice versa, making simplifications and proofs much more manageable.

Cosine Function

The cosine function is one of the primary trigonometric functions. It expresses the ratio of the adjacent side to the hypotenuse in a right-angle triangle. The cosine of angle \( \theta \), denoted as \( \cos \theta \), is crucial in trigonometry and appears frequently in various identities and equations.

• The range of cosine is between -1 and 1.
• \( \cos \theta \) is an even function, meaning \( \cos(-\theta) = \cos \theta \).

In our exercise, \( \cos \theta \) is at the core of the identity we are proving. We use it both in its original form and as part of the Pythagorean identity transformation. When solving trigonometric identities, recognizing the properties and behaviors of functions like cosine helps us manipulate and simplify expressions. Understanding the geometric interpretation—that it represents the horizontal coordinate in the unit circle—also offers insight into its transformations and uses in trigonometry.

Sine Function

Another integral component of trigonometry is the sine function. It gives us the ratio of the opposite side to the hypotenuse in a right triangle for a given angle \( \theta \). Represented as \( \sin \theta \), this function plays a central role in trigonometric identities and equations.

• The range of \( \sin \theta \) is also between -1 and 1.
• \( \sin \theta \) is an odd function, meaning \( \sin(-\theta) = -\sin \theta \).

In the context of the original problem, the sine function is transformed using the Pythagorean identity. By substituting \( \sin^2 \theta \) with \( 1 - \cos^2 \theta \), we demonstrate how these primary trigonometric functions relate and can be manipulated algebraically. This is instrumental in solving complex identities by breaking them down into simpler components using known identities. The sine function, with its distinct properties and its relation to the unit circle as a vertical component, is a fundamental element in these transformations and simplifications.

Algebraic Manipulation

Algebraic manipulation is an essential skill in working with trigonometric identities. It involves transforming expressions through operations such as factoring, substituting, expanding, and simplifying to achieve a desired form or solution.

• In finding identities, algebraic manipulation often involves canceling out terms.
• It's about restructuring expressions without changing their inherent values.

In the original solution, algebraic manipulation is used to factor the numerator and cancel terms in the fraction \( \frac{1 - \cos^2 \theta}{1 + \cos \theta} \). By factoring \( 1 - \cos^2 \theta \) into \( (1 - \cos \theta)(1 + \cos \theta) \), we can simplify the fraction by removing the common terms. This manipulation is key to demonstrating the equivalence of the two sides of the equation, thereby proving the identity. Effective algebra involves recognizing these opportunities for simplification and applying identities wisely, thus reducing complex expressions to simpler forms.