Problem 86 - Contemporary Precalculus

Question

Prove the identity. $$\frac{\cos 8 x+\cos 4 x}{\cos 8 x-\cos 4 x}=-\cot 6 x \cot 2 x$$

Step-by-Step Solution

Answer

Question: Prove that $\frac{\cos 8x + \cos 4x}{\cos 8x - \cos 4x} = -\cot 6x \cot 2x$. Answer: To prove this identity, we have used sum-to-product identities, double-angle formulas, and the reciprocal identities, resulting in the expression $-\cot 6x \cot 2x$.

Step 1: Rewrite the left side using sum-to-product identities

Let's use the sum-to-product identities to rewrite the left side of the equation. The sum-to-product identities are given by: $$\cos A + \cos B = 2\cos \frac{A+B}{2}\cos \frac{A-B}{2}$$ $$\cos A - \cos B = -2\sin \frac{A+B}{2}\sin \frac{A-B}{2}$$ Applying these identities to our problem, we get: $$\frac{\cos 8x + \cos 4x}{\cos 8x - \cos 4x} = \frac{2\cos(\frac{12x}{2})\cos(\frac{4x}{2})}{-2\sin(\frac{12x}{2})\sin(\frac{-4x}{2})}$$

Step 2: Simplify the expression

Now, let's simplify the expression obtained in Step 1: $$\frac{2\cos(6x)\cos(2x)}{-2\sin(6x)\sin(-2x)}$$ Since \(-\sin(-2x) = \sin(2x)\), we can rewrite the expression as: $$\frac{2\cos(6x)\cos(2x)}{2\sin(6x)\sin(2x)}$$

Step 3: Use the reciprocal identities

Finally, let's use the reciprocal identities to replace the sine functions, since \(\cot x\) is defined as \(\frac{\cos x}{\sin x}\): $$\frac{\cos(6x)\cos(2x)}{\sin(6x)\sin(2x)} = \left(\frac{\cos(6x)}{\sin(6x)}\right) \left(\frac{\cos(2x)}{\sin(2x)}\right) = \cot(6x) \cot(2x)$$ Hence, we have proved that: $$\frac{\cos 8x + \cos 4x}{\cos 8x - \cos 4x} = -\cot 6x \cot 2x$$

Key Concepts

Trigonometric Identities

Trigonometric identities are mathematical equations that relate the trigonometric functions of angles. They are used extensively in various fields, including physics, engineering, and mathematics, to solve problems involving triangles and periodic phenomena.

One set of these identities are the sum-to-product identities, which are particularly useful in simplifying the expressions involving sums or differences of cosine or sine functions. An example of these is the identity used in the exercise, where the sum and difference of \(\cos A\) and \(\cos B\) are expressed in terms of the product of cosines. These transformation techniques are not only valuable for mathematical proofs but also instrumental in solving integration problems and analyzing wave interference in physics.

Understanding and applying these identities require a solid grasp of the basic trigonometric functions and their properties. By leveraging these relationships, complex trigonometric expressions can be converted into simpler forms that are easier to analyze or solve.

Cotangent Function

The cotangent function, denoted as \(\cot x\), is a fundamental trigonometric function defined as the reciprocal of the tangent. Specifically, it is the ratio of the adjacent side to the opposite side in a right-angled triangle, or equivalently, the ratio of the cosine function to the sine function: \(\cot x = \frac{\cos x}{\sin x}\).

This definition fits perfectly within the framework of reciprocal identities, which is a class of trigonometric identities. In our exercise, the cotangent function emerges naturally when we express the given ratio in terms of \(\cot\). Understanding the cotangent function is crucial as it frequently appears in trigonometric equations, calculus, and even in the analysis of periodic processes such as waves and oscillations. Appreciating how it relates to the other trigonometric functions can significantly enhance a student's problem-solving abilities.

Reciprocal Identities

Reciprocal identities are a group of trigonometric identities that illustrate the reciprocal relationship between the sine, cosine, tangent, cotangent, secant, and cosecant functions. These identities are expressed as: \(\sin x = \frac{1}{\csc x}\), \(\cos x = \frac{1}{\sec x}\), and \(\tan x = \frac{1}{\cot x}\), among others.

The significance of reciprocal identities is highlighted in the step-by-step exercise, where transforming \(\sin\) terms into \(\cot\) terms simplifies the given expression and leads to the proof of the identity. Recognizing how these identities work provides the foundational knowledge needed to manipulate and simplify complex trigonometric expressions. It's an essential skill for students to grasp as it is widely applicable in solving trigonometric equations and in understanding the interconnections between different trigonometric functions.