Problem 66 - Algebra and Trigonometry Real Mathematics, Real People

Question

Verify the identity. $$\cos (x+y)+\cos (x-y)=2 \cos x \cos y$$

Step-by-Step Solution

Answer

The given identity is verified as true. The sum of cosines of the sum and difference of x and y is equal to twice the product of the cosines of x and y.

Step 1: Apply the sum and difference formulas

Break down \(\cos(x+y)\) and \(\cos(x-y)\) separately using the sum and difference formulas for cosine. The application gives us \(\cos(x + y) = \cos x \cos y - \sin x \sin y\) and \(\cos(x - y) = \cos x \cos y + \sin x \sin y\).

Step 2: Add the equations

Summing up \(\cos(x + y)\) and \(\cos(x - y)\) gives \(\cos(x + y) + \cos(x - y) = \cos x \cos y - \sin x \sin y + \cos x \cos y + \sin x \sin y\). This simplifies to \(\cos(x + y) + \cos(x - y) = 2 \cos x \cos y\).

Step 3: Conclusion

The result after adding the equations confirms the given identity, \(\cos (x+y)+\cos (x-y)=2 \cos x \cos y\).

Key Concepts

Cosine Sum and Difference Formulas

Understanding the cosine sum and difference formulas is essential for solving a variety of trigonometry problems. These formulas express the cosine of the sum \textbf{(x+y)} or difference \textbf{(x-y)} of two angles in terms of the sines and cosines of the individual angles. The sum formula states:
\[ \text{cos}(x + y) = \text{cos} x \text{ cos} y - \text{sin} x \text{ sin} y \]
The difference formula, on the other hand, is:
\[ \text{cos}(x - y) = \text{cos} x \text{ cos} y + \text{sin} x \text{ sin} y \]
When it comes to applying these formulas in problem solving, make sure you correctly identify which formula to use based on the sign between the angles. Remember that these identities are true for all values of x and y. Through practice, they become powerful tools, allowing you to manipulate trigonometric expressions and solve complex equations.

Verifying Trigonometric Identities

Trigonometric identities are equations that are true for all values within the domains of the variables. Verifying these identities often involves a series of algebraic manipulations. It's crucial to understand that verifying an identity means showing that both sides of the equation are equivalent for all possible values of the variables involved.

• Begin by working with one side of the identity to transform it into the other side.
• Look for opportunities to apply fundamental trigonometric identities, such as the Pythagorean identities or the sum and difference formulas.
• Simplify complex expressions by combining like terms and factoring when possible.
• Always keep in mind that the goal is to make both sides of the equation look exactly the same.

For example, when verifying the identity \[ \text{cos}(x+y) + \text{cos}(x-y) = 2 \text{cos} x \text{ cos} y \],we apply the sum and difference formulas as the key step to show equivalence between the two sides.

Trigonometry Problem Solving

Solving trigonometry problems is not just about knowing formulas, but also about applying them in the correct context. When faced with a complex problem, here are some strategies to consider:

• Break the problem into smaller parts and analyze each piece.
• Draw diagrams where possible to visualize the relationships between angles and sides.
• Use algebraic techniques to rearrange equations and isolate variables.
• Apply appropriate trigonometric identities to simplify expressions.
• Check your answers by substituting them back into the original equation or by using a different method to verify results.

In the exercise example of verifying an identity, problem-solving involves recognizing that sum and difference identities are applicable and then methodically simplifying the expressions to demonstrate that both sides of the identity are equivalent.