Problem 41 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 37-46: Verify the identity. $$ \cos (u+v)+\cos (u-v)=2 \cos u \cos v $$

Step-by-Step Solution

Answer

The identity is verified using the sum-to-product identities.

Step 1: Use the sum-to-product identities

The identity we want to verify is \( \cos(u+v) + \cos(u-v) = 2 \cos u \cos v \). From trigonometry, the sum-to-product identity for cosines states \( \cos A + \cos B = 2 \cos \left( \frac{A+B}{2} \right) \cos \left( \frac{A-B}{2} \right) \). Substitute \( A = u+v \) and \( B = u-v \) into the formula.

Step 2: Apply substitution in the identity

Substitute the values: \( A = u+v \) and \( B = u-v \) into the sum-to-product identity: \( \cos(u+v) + \cos(u-v) = 2 \cos \left( \frac{(u+v) + (u-v)}{2} \right) \cos \left( \frac{(u+v) - (u-v)}{2} \right) \).

Step 3: Simplify the expression

Simplify the terms in the expression: \( \frac{(u+v) + (u-v)}{2} = \frac{2u}{2} = u \) and \( \frac{(u+v) - (u-v)}{2} = \frac{2v}{2} = v \).

Step 4: Conclude with the identity

Substituting the simplified terms back, the identity becomes: \( 2 \cos(u) \cos(v) \). The original statement \( \cos(u+v) + \cos(u-v) = 2 \cos u \cos v \) holds true.

Key Concepts

Sum-to-Product Identities

Sum-to-product identities in trigonometry are handy for transforming sums of trigonometric functions into products. They simplify expressions and can solve trigonometric equations more easily. A sum-to-product identity for cosine states:
\[ \cos A + \cos B = 2 \cos \left( \frac{A+B}{2} \right) \cos \left( \frac{A-B}{2} \right) \]
This identity is particularly useful because it reduces addition, which might be more cumbersome, into multiplication, which is often simpler to work with.

• In our given problem, we utilize this identity by substituting specific values into \(A\) and \(B\).
• For instance, if \(A = u+v\) and \(B = u-v\), we plug these into the sum-to-product identity formula.

This strategy converts a sum of cosines into a product of cosines, aiding in simplification.

Cosine Addition Formula

The cosine addition formula is a key player in the toolbox of trigonometric identities. It is used to find the cosine of an angle that is the sum of two other angles, and it's defined as:
\[ \cos(u+v) = \cos u \cos v - \sin u \sin v \]
For our exercise, we're actually looking at the dual usage of this kind of formula. Specifically, it assists in verifying identities, like turning the sum of angles back into simpler terms. However, in our case, we focus on a different but related identity that helps bridge sections of an equation into a more manageable expression.

• This formula comes in handy for complex expressions, enabling us to track and simplify what would otherwise seem intricate.
• It gives us guidance for manipulating problems to discover deeper equivalences.

In essence, understanding and applying the cosine addition formula lays the groundwork for comprehending more complicated trigonometric processes.

Trigonometry Simplification

Simplification in trigonometry involves reducing complex expressions into more straightforward, usable forms. It can be done through identities, algebraic manipulation, and strategic substitutions.
For the problem at hand, the act of simplification is evident when:

• We substitute specific values into the sum-to-product identity to usher in simplification.
• We focus on reducing the compound angles \( u+v \) and \( u-v \) to simpler terms \( u \) and \( v \) respectively.

The follow-up simplification step ensures our result is tidy and prepared for real-world application or further analytical work.
The critical goal is to understand that what starts as a potentially intimidating expression can become clear and manageable when analytical simplification is applied correctly.This approach underpins most strategies in trigonometry to unlock simpler solutions from complex equations.