Problem 64 - Contemporary Precalculus

Question

Determine graphically whether the equa. tion could possibly be an identity. If it could, prove that it is. $$\cos 8 x=\cos ^{2} 4 x-\sin ^{2} 4 x$$

Step-by-Step Solution

Answer

Answer: Yes, the given equation is an identity.

Step1: Simplify the given equation using the double angle formula for cosine

We begin by applying the double angle formula for cosine. In this case, \(\theta = 4x\). From \(\cos 2 \theta = 2\cos^{2}\theta - 1\), we get $$\cos 8x = 2\cos^{2}4x - 1$$ From \(\cos 2 \theta = 1-2\sin^{2}\theta\), we get $$\cos 8x = 1 - 2\sin^{2}4x$$ Now, let's compare this with the given equation: $$\cos 8 x=\cos ^{2} 4 x-\sin ^{2} 4 x$$

Step2: Find the relationship between \(\cos^2 4x\) and \(1 - 2\sin^2 4x\)

We know that $$\cos^{2} \theta + \sin^{2} \theta = 1$$ If we replace \(\theta\) with \(4x\), we get $$\cos ^{2} 4 x + \sin^{2} 4x = 1$$ Now, we can rewrite this equation in terms of \(\cos^2 4x\): $$\cos^{2} 4x = 1 - \sin^{2} 4x$$

Step3: Substitute the derived expression into the given equation and verify the identity

Now let's substitute the expression we derived for \(\cos^2 4x\) back into the original equation and see if both sides are equal: $$\cos 8x = \cos^2 4x - \sin^2 4x$$ $$\cos 8x = (1-\sin^2 4x) - \sin^2 4x$$ Now, remember that we found two expressions for \(\cos 8x\) in Step 1: $$\cos 8x = 2\cos^{2} 4x - 1$$ $$\cos 8x = 1 - 2\sin^{2} 4x$$ Replacing \(\cos 8x\) with the second expression from Step 1 gives us: $$1 - 2\sin^{2} 4x = (1-\sin^2 4x) - \sin^2 4x$$ Now both sides of the equation match, meaning this equation is indeed an identity: $$\cos 8 x=\cos ^{2} 4 x-\sin ^{2} 4 x$$

Key Concepts

Double Angle Formula

The double angle formulas are a set of trigonometric identities that relate the sine and cosine functions of double angles to those of single angles. These formulas are immensely useful in various areas of mathematics including geometry and calculus, as well as in physics and engineering problems involving oscillations and waves.

• \textbf{Cosine Double Angle Formula:} The formula for cosine is expressed as \[ \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \] which can also be transformed into two other equivalent forms: \[ \cos(2\theta) = 2\cos^2(\theta) - 1 \] or \[ \cos(2\theta) = 1 - 2\sin^2(\theta) \]
• \textbf{Sine Double Angle Formula:} Similarly, for the sine function, the double angle formula is \[ \sin(2\theta) = 2\sin(\theta)\cos(\theta) \]

For a student trying to understand and remember these, visual aids such as unit circle diagrams and graphing calculators can be helpful. By comparing the original exercise with these formulas, it becomes clear how to begin simplifying the given trigonometric identity equation and ultimately prove it.

Cosine Function

The cosine function is one of the primary trigonometric functions and is abbreviated as \( \cos \). It describes the ratio of the adjacent side to the hypotenuse of a right-angled triangle. In the unit circle, it represents the x-coordinate of a point determined by an angle originated from the positive x-axis.

Some key properties of the cosine function include:

• It is an even function, which means that \( \cos(\theta) = \cos(-\theta) \).
• The function has a range of [-1, 1], as no side of a triangle can have a length greater than the hypotenuse.
• It has a period of \( 2\pi \), meaning the function repeats its values every \( 2\pi \) radians.

Understanding the cosine function is crucial when learning how to solve trigonometric equations and identities. In the given exercise, the cosine function's properties are used to manipulate the equation to verify the identity.

Sine Function

The sine function, denoted as \( \sin \), is another foundational trigonometric function. It gives the ratio of the opposite side to the hypotenuse in a right-angle triangle. On the unit circle, it reflects the y-coordinate of a point corresponding to an angle traced from the positive x-axis.

Here are several important aspects of the sine function:

• The sine function is an odd function, meaning that \( \sin(-\theta) = -\sin(\theta) \).
• Its values also lie between -1 and 1, paralleling the cosine function, as the height of a triangle cannot exceed the length of its hypotenuse.
• It has a period of \( 2\pi \), similarly to cosine, implying its values repeat after every \( 2\pi \) radians.

The sine function is integral to solving problems involving periodic phenomena like waves. In our textbook problem, understanding the sine function allows us to rearrange the equation efficiently and prove the given trigonometric identity.