Problem 87 - A Graphical Approach to Precalculus with Limits

Question

Verify that equation is an identity. \(\frac{\sin ^{2} \gamma}{\cos \gamma}=\sec \gamma-\cos \gamma\)

Step-by-Step Solution

Answer

The equation is an identity as both sides are equal.

Step 1: Expand the left side

Start with the left side of the equation, \[ \frac{\sin^{2} \gamma}{\cos \gamma}. \] We can rewrite \( \sin^{2} \gamma \) using the identity \( \sin^{2} \gamma = 1 - \cos^{2} \gamma \). Thus, the expression becomes:\[ \frac{1 - \cos^{2} \gamma}{\cos \gamma}. \]

Step 2: Simplify the expression

Next, simplify the fraction \( \frac{1 - \cos^{2} \gamma}{\cos \gamma} \).This simplifies as:\[ \frac{1}{\cos \gamma} - \frac{\cos^{2} \gamma}{\cos \gamma} = \sec \gamma - \cos \gamma. \]

Step 3: Compare with the right side

Now, look at the simplified expression from the left side: \[ \sec \gamma - \cos \gamma \] This matches the right side of the given equation. Therefore, both sides are equal which verifies that the given equation is an identity.

Key Concepts

Trigonometric Functions

Trigonometric functions like sine (\( \sin \)), cosine (\( \cos \)), and secant (\( \sec \)) are fundamental in exploring the relationships between the angles and sides of triangles, especially right triangles. These functions help convert geometric problems into different mathematical forms.

Key trigonometric functions include:

• Sine (\( \sin \)): represents the ratio of the opposite side to the hypotenuse in a right triangle. It's often used to express vertical components of a vector.
• Cosine (\( \cos \)): denotes the ratio of the adjacent side to the hypotenuse. It usually describes horizontal components of a vector.
• Secant (\( \sec \)): is the reciprocal of cosine, or \( \sec \theta = \frac{1}{\cos \theta} \). This function emerges in various mathematical contexts, facilitating reciprocal calculations.

Understanding how these functions interrelate is crucial. They allow us to transform and manipulate mathematical expressions to show equivalence or solve problems. Recognizing standard identities such as \( \sin^2 \gamma + \cos^2 \gamma = 1 \) or using reciprocal identities like \( \sec \gamma = \frac{1}{\cos \gamma} \) can simplify or solve trigonometric equations with ease.

Simplifying Expressions

Simplifying expressions in trigonometry involves rewriting a complex or lengthy expression in a more straightforward or recognizable form. This process is essential for easy comparison and verification of identities.

The key steps in simplifying trigonometric expressions involve:

• Applying identities: Use known identities, such as \( \sin^2 \gamma = 1 - \cos^2 \gamma \), to transform parts of the expression. These identities enable conversion between different trigonometric functions.
• Simplifying fractions: Break down fractions into simpler parts. For example, \( \frac{1 - \cos^2 \gamma}{\cos \gamma} \) can be split into \( \frac{1}{\cos \gamma} - \frac{\cos^2 \gamma}{\cos \gamma} \).
• Combining like terms: After applying identities and simplifying fractions, ensure all similar terms are combined, making the comparison to another expression or equation more straightforward.

Consideration of these steps greatly aids in verifying trigonometric identities. By expressing both sides of an equation in as simple terms as possible, you can more easily assess their equality.

Verifying Identities

Verifying trigonometric identities involves proving that two expressions are equivalent for any angle involved. It requires systematic manipulation of one or both sides of the given equation using trigonometric identities.

The process involves:

• Expanding sides of the equation: Start with the more complex side and apply fundamental identities to transform it. For example, rewriting \( \sin^2 \gamma \) as \( 1 - \cos^2 \gamma \) can provide simpler comparisons.
• Simplifying both sides: Use identities and algebraic manipulation like factoring or combining fractions. Once simplified, ensure both expressions look identical.
• Constant comparison: Compare the results after each simplification step. In our exercise, transforming \( \frac{\sin^2 \gamma}{\cos \gamma} \) into \( \sec \gamma - \cos \gamma \) validated their equivalency.

Handling these steps with clarity and precision can confirm if the equation holds true, effectively verifying the identity. Practicing this method cultivates a deeper understanding of trigonometric relationships and strengthens algebraic skills.