Problem 40 - Algebra and Trigonometry

Question

Use one or more of the six sum and difference identities to solve Exercises \(13-54\) Verify each identity. $$ \cos (\alpha+\beta)+\cos (\alpha-\beta)=2 \cos \alpha \cos \beta $$

Step-by-Step Solution

Answer

The left-hand side of the equation simplifies to \( 2 \cos \alpha \cos \beta \), which is the same as the right-hand side, verifying the identity true.

Step 1: Break the Equation Down

Start by looking at the problem. The equation given is: \( \cos(\alpha+\beta)+\cos(\alpha-\beta) = 2 \cos \alpha \cos \beta \). The task is to prove this identity. It will be proven by separately formulating both left hand side and right hand side of the equation using the cosines of sum and difference formula.

Step 2: Formulate the Left-Hand Side

The left-hand side of the equation is \( \cos(\alpha+\beta)+\cos(\alpha-\beta) \). Use the cosine of sum and the cosine of difference identities to simplify. The identities are: \( \cos(a+b) = \cos a \cos b - \sin a \sin b \) and \( \cos (a-b) = \cos a \cos b + \sin a \sin b \). If these identities are applied to the left-hand side, it becomes \( \cos \alpha \cos \beta - \sin \alpha \sin \beta + \cos \alpha \cos \beta + \sin \alpha \sin \beta \). This simplifies to \( 2 \cos \alpha \cos \beta \), which is the left-hand side of the equation.

Step 3: Comparing Left and Right-Hand Sides

The right-hand side of the equation is \( 2 \cos \alpha \cos \beta \) which, moreover, was ideally shown to be the same as our left-hand side. Therefore, the original identity is verified true.

Key Concepts

Cosine Sum Formula

The cosine sum formula is a fundamental trigonometric identity used to simplify expressions involving the cosine of two added angles. It is expressed as:\[ \cos(a + b) = \cos a \cos b - \sin a \sin b \]This identity allows us to express the cosine of a sum of two angles in terms of the sines and cosines of the individual angles. By applying this formula, you can easily break down a complex cosine expression into simpler components that are easier to handle.

• Usage in Problems: The cosine sum formula is widely used to simplify trigonometric equations and identities — particularly useful in verifying identities or solving equations.
• Importance for Calculations: When dealing with angles that are not standard (like 45°, 60°, or multiples of these angles), the cosine sum formula helps convert expressions into known values.

When working through exercises, it's crucial to remember this formula as it serves as a basis for understanding more complex trigonometric manipulations. In the given exercise, it helps verify the identity by simplifying the left-hand side.

Cosine Difference Formula

The cosine difference formula is another vital trigonometric identity, closely related to the cosine sum formula. It is used to account for the cosine of the difference between two angles. Mathematically, it is given by:\[ \cos(a - b) = \cos a \cos b + \sin a \sin b \]This formula, like its sum counterpart, allows you to decompose the expression into sines and cosines of the individual angles, but importantly, with a plus sign between the sine terms.

• Role in Solving Problems: The cosine difference formula is essential in verifying identities and solving problems where the angles are given as a subtraction.
• Application Example: In the identity verification exercise, it simplifies the sum of cosines of the difference and sum, aiding in achieving an equivalent form across the equation

Knowing the cosine difference formula is crucial for simplifying complex trigonometric expressions. Combining the sum and difference formulas provides a comprehensive approach to handling various angle configurations in trigonometry.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for any value of the involved variables. They play an essential role in simplifying expressions, proving statements, and solving trigonometric equations.

• Types: These identities include basic ones like Pythagorean identities, and more complex ones such as sum and difference identities, double-angle, and half-angle identities.
• Importance: Understanding and using these identities is foundational in trigonometry. They allow mathematicians and students alike to transform complex expressions into simpler or more workable forms.

In the given exercise, the use of the cosine sum and difference identities exemplifies the practical application of trigonometric identities. The ability to manipulate and verify expressions rests heavily on your comfort with these fundamental identities. Mastery of trigonometric identities is not only crucial for success in trigonometry but also in applying these concepts across calculus and various applications in physics and engineering where angles and oscillations are studied.