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Magazine Chapter 19: Problem 1865
\(\cos (x-y)=a, \cos (x+y) \Rightarrow \cot x \cot y=\) (a) \([(a-1) /(a+1)]\) (b) \([(a+1) /(a-1)]\) (c) \(a-1\) (d) \(a+1\)
Short Answer
Expert verified
The short answer is: \(\cot{x} \cot{y} = \frac{(a+1)}{(a-1)}\).
Step by step solution
01
Write down useful trigonometric identities
In this problem, we will need to use two product-to-sum trigonometric identities:
1. \(\cos{A} \cos{B} = \frac{1}{2}[\cos(A-B) + \cos(A+B)]\)
2. \(\sin{A} \sin{B} = \frac{1}{2}[\cos(A-B) - \cos(A+B)]\)
We will also need the relationships of cotangent with cosine and sine:
3. \(\cot{A} = \frac{\cos{A}}{\sin{A}}\)
02
Express cotangent in terms of cosine and sine
Write down the expression for \(\cot{x} \cot{y}\) using relationship 3:
\[\cot{x} \cot{y} = \frac{\cos{x}\cos{y}}{\sin{x}\sin{y}}\]
03
Use product-to-sum formulas
Substitute the product expressions for the cosines and sines in the numerator and denominator of the cotangent expression using the product-to-sum identities 1 and 2:
\[\cot{x} \cot{y} = \frac{\frac{1}{2}[\cos(x-y) + \cos(x+y)]}{\frac{1}{2}[\cos(x-y) - \cos(x+y)]}\]
04
Cancel out common terms and simplify
Cancel out the common term \(\frac{1}{2}\) in the numerator and denominator, and substitute the given value of \(\cos(x-y) = a\) in the expression:
\[\cot{x} \cot{y} = \frac{a + \cos(x+y)}{a - \cos(x+y)}\]
05
Compare with the given options
Compare the simplified expression with the given options. The expression obtained in Step 4 matches option (b):
\(\cot{x} \cot{y} = [(a+1)/(a-1)]\)
Therefore, the correct option is (b).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
product-to-sum identities
Product-to-sum identities are essential tools in trigonometry. These identities transform the product of trigonometric functions into a sum or difference, making complex expressions easier to manage. This transformation is handy in integration and simplifying expressions, as well as solving equations.
To get a better grip, here are two essential product-to-sum identities:
To get a better grip, here are two essential product-to-sum identities:
- For cosine: \[\cos{A} \cos{B} = \frac{1}{2}[\cos(A-B) + \cos(A+B)]\]
- For sine: \[\sin{A} \sin{B} = \frac{1}{2}[\cos(A-B) - \cos(A+B)]\]
cosine formulas
Cosine formulas are fundamental when studying relationships between angles and sides in mathematics. Central to our problem is the expression for cosine differences: \[\cos(x-y) = a\]This formula helps us further deduce relationships using given or known values.
When working with various cosine expressions, like \(\cos(x+y)\), understanding their behavior enables correct substitution into identities or transformations. Notably, cosine values range from -1 to 1, making them pivotal in understanding the symmetric aspects of trigonometric graphs. In problem-solving, proper manipulation of such expressions will often lead to simplifying otherwise complex or challenging problems.
When working with various cosine expressions, like \(\cos(x+y)\), understanding their behavior enables correct substitution into identities or transformations. Notably, cosine values range from -1 to 1, making them pivotal in understanding the symmetric aspects of trigonometric graphs. In problem-solving, proper manipulation of such expressions will often lead to simplifying otherwise complex or challenging problems.
cotangent expression
Cotangent is another trigonometric function related to cosine and sine, defined as:\[\cot{A} = \frac{\cos{A}}{\sin{A}}\]This means the cotangent describes the ratio of the adjacent side to the opposite side in a right-angled triangle. Here, when dealing with two angles \(x\) and \(y\), their interaction is expressed through:\[\cot{x} \cot{y} = \frac{\cos{x}\cos{y}}{\sin{x}\sin{y}}\]This expression can further be expanded or simplified using product-to-sum identities, making it manageable to interpret complex trigonometric expressions. Understanding cotangent in this form helps demystify how angles can be combined and manipulated in a broad array of mathematical contexts.
trigonometric transformations
Trigonometric transformations allow us to change expressions into more usable forms, whether it's simplifying or solving equations. One such transformation is using product-to-sum identities to convert products into a sum or difference. In this context, we took:\[\cot{x} \cot{y} = \frac{\frac{1}{2}[\cos(x-y) + \cos(x+y)]}{\frac{1}{2}[\cos(x-y) - \cos(x+y)]}\]and simplified it by canceling out common terms, yielding a more straightforward expression.
Transformations are crucial in trigonometry for:
Transformations are crucial in trigonometry for:
- Simplifying complex equations.
- Solving systems of equations where initial forms are too cumbersome.
- Enabling the application of known identities and properties efficiently.
