Problem 90 - Precalculus Mathematics for Calculus

Question

Prove the identity. $$\tan y=\frac{\sin (x+y)-\sin (x-y)}{\cos (x+y)+\cos (x-y)}$$

Step-by-Step Solution

Answer

The identity is proven true as both sides simplify to \( \tan y \).

Step 1: Recall the Trigonometric Identities

We start by recalling the sum and difference identities for sine and cosine. Specifically: \( \sin(a + b) = \sin a \cos b + \cos a \sin b \) and \( \sin(a - b) = \sin a \cos b - \cos a \sin b \). Also for cosine: \( \cos(a + b) = \cos a \cos b - \sin a \sin b \) and \( \cos(a - b) = \cos a \cos b + \sin a \sin b \).

Step 2: Apply Trigonometric Identities to the Numerator

Calculate the expression \( \sin (x+y) - \sin (x-y) \) using the identities:\[ \sin(x+y) - \sin(x-y) = (\sin x \cos y + \cos x \sin y) - (\sin x \cos y - \cos x \sin y) \]Simplifying gives:\[ \sin(x+y) - \sin(x-y) = 2 \cos x \sin y \]

Step 3: Apply Trigonometric Identities to the Denominator

Now, calculate \( \cos (x+y) + \cos (x-y) \) using the identities:\[ \cos(x+y) + \cos(x-y) = (\cos x \cos y - \sin x \sin y) + (\cos x \cos y + \sin x \sin y) \]Simplifying gives:\[ \cos(x+y) + \cos(x-y) = 2 \cos x \cos y \]

Step 4: Substitute Simplified Numerator and Denominator

Substitute the results from Step 2 and Step 3 back into the expression for \( \tan y \): \[ \tan y = \frac{2 \cos x \sin y}{2 \cos x \cos y} \]Simplify the expression by cancelling \( 2 \cos x \) from the numerator and the denominator:

Step 5: Simplify the Expression

After cancellation, we obtain:\[ \tan y = \frac{\sin y}{\cos y} \]This is the definition of the tangent function, therefore:\[ \tan y = \tan y \]

Step 6: Conclusion

The identity \( \tan y=\frac{\sin(x+y)-\sin(x-y)}{\cos(x+y)+\cos(x-y)} \) is proven to be true by showing that the right-hand side simplifies to \( \tan y \).

Key Concepts

Tangent Function

The tangent function is a fundamental concept in trigonometry used to describe the ratio between the lengths of two sides of a right-angled triangle. Specifically, it is the ratio of the opposite side to the adjacent side for a given angle. This can also be expressed using the sine and cosine functions:

• The function is defined as: \( \tan y = \frac{\sin y}{\cos y} \).
• It's periodic, meaning that it repeats its values every \( \pi \) radians.
• When plotted on a graph, \( \tan y \) has vertical asymptotes at odd multiples of \( \frac{\pi}{2} \).

The significance of the tangent function in our exercise lies in demonstrating the relationship between a complex trigonometric expression and the basic identity of tangent itself. This exercise showcases how understanding and manipulating identities can be powerful in reducing more involved expressions to simpler, familiar forms.

Sine and Cosine

Sine and cosine are the basic building blocks of trigonometry. Both are crucial in defining other trigonometric functions and identities. Let's break down their properties and roles:

• Sine (\( \sin \)) represents the y-coordinate of a point on the unit circle at a certain angle.
• Cosine (\( \cos \)) gives the x-coordinate of that point on the unit circle.
• Both functions oscillate between -1 and 1, sharing a period of \( 2\pi \) radians.
• The sum and difference identities of sine and cosine are powerful tools: \( \sin(a \pm b) = \sin a \cos b \pm \cos a \sin b \) \( \cos(a \pm b) = \cos a \cos b \mp \sin a \sin b \)

In the provided proof, the identities simplify the original trigonometric expressions. Simplifying such expressions using these identities is a common practice in solving trigonometric equations and proving identities, as it often reveals underlying simpler structures.

Proving Identities

Proving trigonometric identities is a skill that builds deep understanding of trigonometric relationships and improves problem-solving abilities. Here's a mini-guide to approach:

• Understand the Given Identity: Comprehend what the problem demands. Here, the goal is to prove a given expression equals \( \tan y \).
• Simplify the Expression: Utilize known identities like sum and difference for sine and cosine to manipulate the expression. This involves substituting parts of the original expression to known identities.
• Show Equivalence: Simplify both sides of the equation step-by-step to show they are equal. For example, in the exercise given, simplifying both the numerator and denominator to mimic the form \( \frac{\sin y}{\cos y} \) was key.
• Verification: Double-check the steps undertaken to ensure they logically lead from the initial statement to the identity being proven.

Throughout this exercise, the overarching trick is to connect complex ratios back to simpler forms that are easily recognizable, such as the basic definition of the tangent function. Practicing this process enhances both algebraic and trigonometric proficiency.