Problem 2 - Algebra 2 and Trigonometry

Question

Cory said that in Example \(3,1-\sin \theta=\frac{\cos ^{2} \theta}{1+\sin \theta}\) could have been shown to be an identity by multiplying the left side by \(\frac{1+\sin \theta}{1+\sin \theta} \cdot\) Do you agree with Cory? Explain why or why not.

Step-by-Step Solution

Answer

Yes, multiplying left side equals right, verifying identity using trigonometric identities.

Step 1: Understand the Given Expression

We want to determine if multiplying the left side of the equation \(1 - \sin \theta\) by \(\frac{1 + \sin \theta}{1 + \sin \theta}\) will turn it into an identity, meaning the left and the right side are equal.

Step 2: Multiply the Left Side

To transform \(1 - \sin \theta\), multiply it by \(\frac{1 + \sin \theta}{1 + \sin \theta}\):\[ \left(1 - \sin \theta\right) \cdot \frac{1 + \sin \theta}{1 + \sin \theta} = \frac{(1 - \sin \theta)(1 + \sin \theta)}{1 + \sin \theta}. \]

Step 3: Apply the Difference of Squares

The numerator \((1 - \sin \theta)(1 + \sin \theta)\) can be expanded using the identity for the difference of squares: \((a - b)(a + b) = a^2 - b^2\). Here, it becomes \(1 - \sin^2 \theta\).

Step 4: Simplify Using Pythagorean Identity

We know from the Pythagorean identity that \(1 - \sin^2 \theta = \cos^2 \theta\). Substitute this back into the expression:\[ \frac{1 - \sin^2 \theta}{1 + \sin \theta} = \frac{\cos^2 \theta}{1 + \sin \theta}. \]

Step 5: Verify Equality

The expression we obtained on the left matches the right side of the original equation. Therefore, multiplying \(1 - \sin \theta\) by \(\frac{1 + \sin \theta}{1 + \sin \theta}\) does indeed show the equation is an identity.

Key Concepts

Understanding the Difference of Squares

The difference of squares is a well-known algebraic identity that allows us to simplify expressions involving products of the form \((a - b)(a + b)\). This identity states that:

• \((a - b)(a + b) = a^2 - b^2\)

In the context of trigonometry, let's apply this to the expression \((1 - \sin \theta)(1 + \sin \theta)\). Here, the variables \(a\) and \(b\) are 1 and \(\sin \theta\), respectively. When we apply the difference of squares formula:

• We get \(1^2 - (\sin \theta)^2\).

This simplifies to \(1 - \sin^2 \theta\), which is crucial for further simplification. Recognizing this pattern is useful, as it breaks down a multiplication into a straightforward difference, making it easier to handle in identities.
It's an essential stepping stone toward applying more complex identities to simplify trigonometric expressions.

The Power of the Pythagorean Identity

In trigonometry, the Pythagorean identity is one of the most important identities, and it expresses a fundamental relationship between the sine and cosine functions. It states that:

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

This identity can be rearranged to help solve different trigonometric problems. For example, by subtracting \(\sin^2 \theta\) from both sides, we get:

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

In our exercise, this rearranged identity allows the expression \(1 - \sin^2 \theta\) to be replaced by \(\cos^2 \theta\).
This is a notable simplification step, as it translates a part of the expression solely in terms of cosine, aligning it directly with the right side of the equation. Therefore, understanding and applying the Pythagorean identity is key to unlocking simplified forms of trigonometric expressions.

Mastering Trigonometric Simplification

Trigonometric simplification involves reducing complex trigonometric expressions to their simplest form. This process often utilizes identities, such as the Pythagorean identity, to make one side of an equation look like the other.
In the given exercise, the goal is to show that \(1 - \sin \theta\) can be transformed to \(\frac{\cos^2 \theta}{1 + \sin \theta}\) by multiplying \(1 - \sin \theta\) by \(\frac{1 + \sin \theta}{1 + \sin \theta}\). This fraction is equivalent to 1, meaning it is essentially multiplying by 1 without changing the value.
By multiplying, we apply the difference of squares and then the Pythagorean identity in succession:

• The numerator becomes \(1 - \sin^2 \theta\) using the difference of squares.
• Then, replace \(1 - \sin^2 \theta\) with \(\cos^2 \theta\) using the Pythagorean identity.

Thus, the expression simplifies to \(\frac{\cos^2 \theta}{1 + \sin \theta}\). This verifies that our initial expression is indeed an identity, demonstrating how these simplification methods clarify trigonometric relationships.