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Magazine Chapter 7: Problem 90
Prove the identity. $$\tan y=\frac{\sin (x+y)-\sin (x-y)}{\cos (x+y)+\cos (x-y)}$$
Short Answer
Expert verified
The identity is proven true as both sides simplify to \( \tan y \).
Step by step solution
01
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 \( \cos(a - b) = \cos a \cos b + \sin a \sin b \).
02
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:\[ \sin(x+y) - \sin(x-y) = 2 \cos x \sin y \]
03
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:\[ \cos(x+y) + \cos(x-y) = 2 \cos x \cos y \]
04
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:
05
Simplify the Expression
After cancellation, we obtain:\[ \tan y = \frac{\sin y}{\cos y} \]This is the definition of the tangent function, therefore:\[ \tan y = \tan y \]
06
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 \( \tan y \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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 \( \frac{\pi}{2} \).
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: \( \sin(a \pm b) = \sin a \cos b \pm \cos a \sin b \) \( \cos(a \pm b) = \cos a \cos b \mp \sin a \sin b \)
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.
