Problem 44 - Precalculus: Functions and Graphs

Question

Exer. \(37-46:\) Verify the identity. $$\cos (u+v) \cdot \cos (u-v)=\cos ^{2} u-\sin ^{2} v$$

Step-by-Step Solution

Answer

The identity is verified: \( \cos(u+v) \cdot \cos(u-v) = \cos^2 u - \sin^2 v \).

Step 1: Understand the Identity

The identity to verify is that \( \cos(u+v) \cdot \cos(u-v) = \cos^2 u - \sin^2 v \). We need to manipulate the left side of the equation to show that it is equal to the right side.

Step 2: Apply the Angle Addition and Subtraction Formulas

Recall the formulas for cosine of sum and difference: \[ \cos(a + b) = \cos a \cos b - \sin a \sin b \] and \[ \cos(a - b) = \cos a \cos b + \sin a \sin b \].We apply these to our expression:\[ \cos(u+v) = \cos u \cos v - \sin u \sin v \] and \[ \cos(u-v) = \cos u \cos v + \sin u \sin v \].

Step 3: Multiply the Formulas

Multiply the two results from Step 2:\[ (\cos u \cos v - \sin u \sin v)(\cos u \cos v + \sin u \sin v) = (\cos u \cos v)^2 - (\sin u \sin v)^2 \].This is based on the difference of squares formula \( (a-b)(a+b) = a^2 - b^2 \).

Step 4: Simplify the Expression

Now we simplify the expression:\[ (\cos u \cos v)^2 - (\sin u \sin v)^2 = \cos^2 u \cos^2 v - \sin^2 u \sin^2 v \].Use the identity \( \cos^2 v + \sin^2 v = 1 \), then recognize that:\[ \cos^2 u \cos^2 v - \sin^2 u \sin^2 v = \cos^2 u(1 - \sin^2 v) - \sin^2 u \sin^2 v \].

Step 5: Final Simplification

Since \( 1 - \sin^2 v = \cos^2 v \) by the Pythagorean identity, the expression reduces to:\[ \cos^2 u - \sin^2 v \].This matches the right side of the original identity.

Key Concepts

Cosine Addition Formula

The cosine addition formula is a fundamental tool in trigonometry. It helps us find the cosine of the sum of two angles. The formula is given by:

• \( \cos(a + b) = \cos a \cos b - \sin a \sin b \)

This formula shows us that the cosine of a sum is not just a simple addition of two cosines. Instead, it combines both the cosine and sine values of the angles involved.
To understand its application, imagine you are given two angles \( u \) and \( v \). To find \( \cos(u+v) \), apply the formula directly by substituting \( a \) and \( b \) with \( u \) and \( v \) respectively.
The cosine addition formula is often used in solving trigonometric identities, such as verifying and simplifying expressions. It is important to remember this formula as it frequently appears in various applications of trigonometry.

Cosine Subtraction Formula

Similar to the cosine addition formula, the cosine subtraction formula helps us find the cosine of the difference between two angles. The formula is:

• \( \cos(a - b) = \cos a \cos b + \sin a \sin b \)

The subtle but important difference here is the plus sign between the sine terms. This signifies that finding cosine for a difference of angles involves adding the product of the sines of the individual angles.
In practice, if you're tasked with finding \( \cos(u-v) \), use this formula. Just replace \( a \) and \( b \) with \( u \) and \( v \).
These trigonometric formulas are indispensable for simplifying expressions where complex angles are involved, and, like the cosine addition formula, it aids in solving problems related to wave functions, oscillations, and periodic phenomena.

Pythagorean Identity

The Pythagorean identity is an anchor in trigonometry, relating the square of sine and cosine of the same angle. It is expressed as:

• \( \cos^2 v + \sin^2 v = 1 \)

This identity is derived from the Pythagorean theorem and states that for any angle \( v \), the sum of the squares of its sine and cosine is always equal to 1.
In the context of verifying trigonometric identities, this identity serves as a powerful tool to simplify complex trigonometric expressions. It often allows for substitution that can connect seemingly unrelated terms.
For instance, in the solution above, \( 1 - \sin^2 v \) was replaced with \( \cos^2 v \) thanks to this identity. Such substitutions can greatly simplify transformations and calculations.

Difference of Squares

The difference of squares is a mathematical concept that is not exclusive to trigonometry but is broadly applied across algebra. It follows the principle:

• \( (a-b)(a+b) = a^2 - b^2 \)

In many trigonometric problems, expressions can be simplified using this rule. By recognizing patterns that fit this form, one can significantly reduce the complexity of an expression.
In the exercise solution provided, the product \( (\cos u \cos v - \sin u \sin v)(\cos u \cos v + \sin u \sin v) \) was simplified using the difference of squares formula to \( (\cos u \cos v)^2 - (\sin u \sin v)^2 \).
Mastering the recognition of this pattern is crucial for both algebraic and trigonometric proficiency, aiding in solving equations efficiently.