Problem 79 - A Graphical Approach to Precalculus with Limits

Question

Write each expression as a product of trigonometric functions or values. $$\cos 4 x-\cos 2 x$$

Step-by-Step Solution

Answer

\(\cos 4x - \cos 2x = -2 \sin(3x) \sin(x)\)

Step 1: Identify the Trigonometric Identity

We need to recognize the trigonometric identity that can simplify the expression \( \cos 4x - \cos 2x \). The suitable identity here is the cosine difference identity: \[ \cos A - \cos B = -2 \sin\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right) \]

Step 2: Assign Variables for the Identity

Assign \(A = 4x\) and \(B = 2x\) in the identity. This helps in simplifying the expression by plugging these values into the identity formula.

Step 3: Calculate the Sum and Difference

Compute \( \frac{A+B}{2} = \frac{4x + 2x}{2} = 3x \) and \( \frac{A-B}{2} = \frac{4x - 2x}{2} = x \). These calculations are required for substituting into the identity.

Step 4: Substitute into the Identity

Use the results from Step 3 in the identity from Step 1: \[ \cos 4x - \cos 2x = -2 \sin(3x) \sin(x) \] This is the expression written as a product of trigonometric functions.

Key Concepts

Cosine Difference

The cosine difference identity is a fundamental tool in trigonometry used to simplify expressions involving the difference of two cosine terms. In the exercise, we started with \( \cos 4x - \cos 2x \), a perfect candidate for applying this identity. The general form of the cosine difference identity is:

• \( \cos A - \cos B = -2 \sin\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right) \)

This identity converts a difference of cosines into a product of sines, simplifying the expression. Understanding this identity is crucial for transforming complex trigonometric expressions into products that are often easier to handle and analyze. By recognizing \( A = 4x \) and \( B = 2x \), we can compute the necessary parts to rewrite the expression using the identity. This essential step paves the way for further simplification through other identities.

Product-to-Sum Identities

Product-to-sum identities are a set of trigonometric identities that convert products of sine and cosine functions into sums or differences. These identities are extremely helpful when dealing with problems that need simplification. One of the main reasons to use the cosine difference identity is to turn subtracted cosine terms into a sine product, which then can be further simplified using product-to-sum identities if necessary.For the expression \(-2 \sin(3x) \sin(x)\), we achieved the simplification from the difference identity. If further transformation is required, product-to-sum identities come into play. For example, the identity for the product of sines is:

• \( 2 \sin A \sin B = \cos(A-B) - \cos(A+B) \)

However, in our exercise, the step of further transforming into product-to-sum is not pursued, as the cosine difference identity already effectively simplifies the original expression. However, knowing these identities is incredibly beneficial for tackling more complex trigonometric problems.

Trigonometric Simplification

Trigonometric simplification refers to the process of reducing complex trigonometric expressions into simpler forms. This often involves transforming sums or differences into products or vice versa, using various identities.In the given exercise, we demonstrated a simplification from the expression \( \cos 4x - \cos 2x \) to \(-2 \sin(3x) \sin(x)\) using trigonometric identities like the cosine difference. This simplification is crucial for solving trigonometric equations, integrating trigonometric functions, or analyzing periodic functions.By simplifying trigonometric expressions:

• Calculations become easier and more straightforward.
• It can reveal periodicity and symmetry in functions.
• It aids in solving complex integrals and derivatives involving trigonometric terms.

Grasping both the basic and advanced identities lets one quickly see which identity might be useful for any given problem. Mastery of these simplifications not only saves time but also enhances the accuracy of mathematical work.