Problem 3 - Algebra 2

Question

Verify that each of the following is an identity. $$ \frac{\cos ^{2} \theta}{1-\sin \theta}=1+\sin \theta $$

Step-by-Step Solution

Answer

The expression is an identity, as both sides simplify to \(1 + \sin \theta\).

Step 1: Understand the Identity

We need to verify that \( \frac{\cos^{2} \theta}{1-\sin \theta} = 1+\sin \theta \). This means we need to show that the expression on the left side is equal to the expression on the right side for all values of \( \theta \) where both sides are defined.

Step 2: Recall Relevant Trigonometric Identity

Remember that one of the basic trigonometric identities is \( \cos^{2} \theta = 1 - \sin^{2} \theta \). This identity will help in simplifying the expression on the left-hand side.

Step 3: Simplify the Left-Hand Side Expression

Start by substituting the identity \( \cos^{2} \theta = 1 - \sin^{2} \theta \) into the left-hand side:\[ \frac{ \cos^{2} \theta }{ 1 - \sin \theta } = \frac{ 1 - \sin^{2} \theta }{ 1 - \sin \theta }. \]

Step 4: Simplify Using Algebraic Manipulation

Now, factor the numerator \(1 - \sin^2 \theta\) as a difference of squares:\[ 1 - \sin^2 \theta = (1 - \sin \theta)(1 + \sin \theta). \]Replace the numerator with the factorization:\[ \frac{ (1 - \sin \theta)(1 + \sin \theta) }{ 1 - \sin \theta }. \]

Step 5: Cancel Common Terms

Notice that \( 1 - \sin \theta \) is both in the numerator and the denominator, so you can cancel them out:\[ \frac{ (1 - \sin \theta)(1 + \sin \theta) }{ 1 - \sin \theta } = 1 + \sin \theta. \]

Step 6: Conclusion

The left-hand side reduces to the right-hand side, so the given expression is an identity. Thus, we have verified that:\[ \frac{\cos^{2}\theta}{1-\sin\theta} = 1+\sin\theta. \]

Key Concepts

Understanding Algebraic Manipulation

Algebraic manipulation is a process that involves using various algebraic techniques to simplify or transform expressions. In the context of trigonometric identities, algebraic manipulation can help us verify if one expression is equal to another by breaking them down into basic identities or simpler forms.

In our exercise, we used algebraic manipulation to simplify the left-hand side of the identity \( \frac{\cos^{2} \theta}{1-\sin \theta} \). By substituting \( \cos^{2} \theta \) with \( 1 - \sin^{2} \theta \), we could break down the expression into more manageable parts. This is an essential step because it allows us to use further simplification strategies, such as recognizing a difference of squares.

Some tips for effective algebraic manipulation include:

• Recognize and apply basic algebraic formulas, like the difference of squares: \( a^2 - b^2 = (a-b)(a+b) \).
• Always look for common factors that can be cancelled out to simplify fractions.
• Keep equations balanced by performing the same operation on both sides when solving for a variable.

Applying the Cosine Identity

The cosine identity is one of the cornerstone identities in trigonometry. The specific identity discussed in our exercise is \( \cos^{2} \theta = 1 - \sin^{2} \theta \). This identity stems from the Pythagorean identity \( \sin^{2} \theta + \cos^{2} \theta = 1 \).

The importance of the cosine identity in the given exercise is that it provides a pathway to transform the expression \( \cos^{2} \theta \) into a format that allows further simplification. By writing \( \cos^{2} \theta \) as \( 1 - \sin^{2} \theta \), we gain the ability to factor it using the difference of squares method.

Understanding how to use the cosine identity is crucial for both simplifying expressions and solving trigonometric equations. Here is a quick tip to keep in mind:

• When you see terms like \( \cos^{2} \theta \), always consider whether substituting it with \( 1 - \sin^{2} \theta \) will make the expression simpler or create a path for further simplification.

Utilizing the Sine Identity

The sine identity \( \sin^{2} \theta + \cos^{2} \theta = 1 \) is often manipulated to express \( \cos^{2} \theta \) or \( \sin^{2} \theta \) in terms of each other. In the context of our exercise, understanding these manipulations is key to verifying trigonometric identities.

In our worked example, while the sine identity itself was not directly manipulated, recognizing the relationship between sine and cosine allowed us to replace \( \cos^{2} \theta \) in terms of sine, leading to a factorable expression. It's critical to recognize when and how to apply such identities.

Here are some handy reminders when dealing with the sine identity:

• If faced with \( \cos^{2} \theta \), think about setting it equal to \( 1 - \sin^{2} \theta \), which can simplify complex expressions.
• Trigonometric identities are interconnected, so keep the relationships \( \sin^{2} \theta + \cos^{2} \theta = 1 \) top of mind, as they often unlock the solution to tricky problems.