Chapter 6: Problem 13
Prove that \(2 \cos x \cos y=\cos (x+y)+\cos (x-y)\)
Short Answer
Expert verified
Verified using the product-to-sum identity: \(2 \, \text{cos} \, x \, \text{cos} \, y = \text{cos} \, (x + y) + \text{cos} \, (x - y)\)
Step by step solution
01
Recall the product-to-sum identities
The product-to-sum identities for trigonometric functions translate a product of sine or cosine into a sum or difference. One such identity that we will use here is: \[2 \, \text{cos} \, A \, \text{cos} \, B = \text{cos} \, (A + B) + \text{cos} \, (A - B).\]
02
Identify the variables
In our given expression, identify \(A = x\) and \(B = y\). Thus, we rewrite the identity replacing \(A\) with \(x\) and \(B\) with \(y\).
03
Apply the identity
Use the identity from step 1, plugging in the values of \(A\) and \(B\):\[ 2 \, \text{cos} \, x \, \text{cos} \, y = \text{cos} \, (x + y) + \text{cos} \, (x - y).\]
04
Comparison
Notice that the left-hand side of the equation is already in the form of \(2 \, \text{cos} \, x \, \text{cos} \, y\). The right-hand side is \(\cos (x + y) + \cos (x - y)\), which matches exactly the right-hand side of the product-to-sum identity stated above. This confirms that the original statement is true.
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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 extremely useful in trigonometry. They help us transform products of sine and cosine functions into sums and differences. This approach simplifies computations and proofs greatly. The key product-to-sum identity is defined as follows: \[ 2 \, \cos \, A \, \cos \, B = \cos \, (A + B) + \cos \, (A - B) \] Here, the product of the cosine functions of two angles can be expressed as the sum of two other cosine functions. This means multiplying two cosines can be turned into adding their combinations, which often makes solving equations and proving identities much easier. Always remember this identity when facing trigonometric expressions in your exercises. Identifying patterns is key to solving complex problems.
Cosine function
The cosine function, denoted as \( \cos \), is one of the fundamental trigonometric functions. It is particularly important in various areas of mathematics and physics due to its properties and applications. The cosine of an angle in a right-angled triangle is defined as the ratio of the length of the adjacent side to the hypotenuse: \[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \] For any angle \( \theta \) on the unit circle, \( \cos \theta \) denotes the horizontal coordinate, simplifying computation and graphical understanding. In our product-to-sum identity, cosine’s ability to combine and simplify products into sums or differences of angles is highly exploited, showcasing its computational versatility.
Trigonometric proofs
Trigonometric proofs involve verifying that two sides of an equation are equivalent by using known identities and properties. For our exercise, we rely on a fundamental product-to-sum identity for cosine. Key steps in solving trigonometric proofs usually include:
- Recognizing patterns and familiar identities
- Substituting appropriate angles
- Rearranging terms to simplify expressions
- Properly applying identities to match both sides of the equation
Precalculus
Precalculus serves as the foundation before diving into calculus. It covers various important mathematical concepts, including trigonometry, which is essential for understanding calculus. Some essential trigonometric concepts in precalculus include:
- Trigonometric functions like sine, cosine, and tangent
- Trigonometric identities, which form the basis for simplifying expressions and proving equations
- Graphing and understanding trigonometric functions
- Solving trigonometric equations