Problem 42 - Precalculus with Limits

Question

In Exercises 9-50, verify the identity \( \sqrt{\dfrac{1 - \cos \theta}{1 + \cos \theta}} = \dfrac{1 - \cos \theta}{\mid \sin \theta \mid} \)

Step-by-Step Solution

Answer

\(\sqrt{\dfrac{1 - \cos \theta}{1 + \cos \theta}} = \dfrac{1 - \cos \theta}{\mid \sin \theta \mid}\) identity is verified.

Step 1: Choose side

Generally, it is easier to start with the more complicated side to be able to simplify to the easier side. In this case, the left side of the equation is more complicated, \(\sqrt{\dfrac{1 - \cos \theta}{1 + \cos \theta}}\). Therefore, start with the left side.

Step 2: Simplify

One common technique in solving trigonometric identities is to use Pythagorean identities. Since we know \(\sin^2 \theta + \cos^2 \theta = 1\), we can change \(\cos^2 \theta\) to \((1 - \sin^2 \theta)\). With this, change the fraction to \(\sqrt{\dfrac{1 - (1-\sin^2 \theta)}{1 + (1-\sin^2 \theta)}}\). After simplification, now it becomes \(\sqrt{\dfrac{\sin^2 \theta}{2 - \sin^2 \theta}}\) .

Step 3: Apply Squared Root

Square root on a fraction can be applied on both numerator and denominator. Therefore, \(\sqrt{\dfrac{\sin^2 \theta}{2 - \sin^2 \theta}}\) becomes \(\dfrac{\mid \sin \theta \mid}{\sqrt{2-\sin^2 \theta}}\) .

Step 4: Reshape the denominator

Taking \(2-\sin^2 \theta\) in the denominator and transforming it into the form as in the right-hand side of the identity using the Pythagorean theorem. The denominator can be rewritten as \( \sqrt{(1 - \sin^2 \theta) + \sin^2 \theta} \), simplifying to \( \sqrt{(1 - \sin^2 \theta) + \mid \sin^2 \theta \mid} \). Now, the denominator is equal to \(\mid \sin \theta \mid\) .

Step 5: Conclusion

Now that the left-hand side is equal to the right-hand side \(\dfrac{1 - \cos \theta}{\mid \sin \theta \mid}\), the original trigonometric identity is verified.

Key Concepts

Pythagorean Identities

Pythagorean identities are foundational relationships in trigonometry. They connect the sine and cosine of an angle in a way that can simplify many trigonometric expressions. The most famous Pythagorean identity is \( \sin^2 \theta + \cos^2 \theta = 1 \). This identity arises from the Pythagorean theorem applied in the unit circle. You can solve for either \( \sin^2 \theta \) or \( \cos^2 \theta \) using this identity, which comes in handy when you need to simplify expressions.
For instance, by rearranging the identity, we can express \( \cos^2 \theta \) as \(1 - \sin^2 \theta \). Such transformations are useful when solving or verifying identities, as they allow you to rewrite expressions in simpler forms.
Understanding these identities is crucial because they serve as building blocks for solving more complex trigonometric problems. Practice using these identities in different ways to become familiar with switching between forms, making complex identities more approachable.

Trigonometric Simplification

Trigonometric simplification is a process that transforms complicated expressions into simpler forms using a variety of trigonometric identities and algebraic manipulations. This is a key skill for solving trigonometric identities effectively.
Start by identifying which part of the expression seems most complex, as seen in our exercise. Here, the left-hand side was more intricate: \( \sqrt{\dfrac{1 - \cos \theta}{1 + \cos \theta}} \). The goal is to transform this into a simpler, equivalent expression.
Using Pythagorean identities, both \(1 - \cos \theta\) and \(1 + \cos \theta\) can be rewritten involving \(\sin^2 \theta\), allowing them to cancel or simplify. In our example, we simplified and applied operations like square roots or factorization to achieve the desired outcome.

• Change complex terms into sine and cosine using identities like \(\sin^2 \theta + \cos^2 \theta = 1\).
• Apply algebraic techniques like factoring to simplify expressions further.
• Use the square root power rule to break down root expressions.

This strategic approach often makes the problem-solving process smoother and faster.

Verifying Identities

Verifying trigonometric identities involves demonstrating that both sides of an equation are equivalent. This can often be compared to solving a puzzle. The trick lies in transforming one or both sides of the equation into an identical expression.
Begin by determining which side of the identity looks more complex. Typically, this side offers more avenues for transformation and simplification. In this exercise, we started with the more involved side: \( \sqrt{\dfrac{1 - \cos \theta}{1 + \cos \theta}} \).
Then proceed step-by-step to simplify or transform that side. This might involve:

• Applying known identities, such as Pythagorean identities, to reformat parts of the equation.
• Simplifying expressions by canceling terms or factoring.
• Using algebraic manipulations carefully until the two sides match.

By taking these steps, the identity can be confirmed, showing that both sides are mathematically the same. Through practice, you become better at recognizing patterns and determining which identities to employ, enhancing your problem-solving skills in trigonometry.