Problem 63 - A Graphical Approach to Precalculus with Limits

Question

Verify that each equation is an identity. $$\cos 2 \theta=\frac{2-\sec ^{2} \theta}{\sec ^{2} \theta}$$

Step-by-Step Solution

Answer

Yes, the equation \( \cos 2 \theta = \frac{2 - \sec^2 \theta}{\sec^2 \theta} \) is an identity.

Step 1: Rewrite the Left Side using a Double Angle Formula

The left side of the equation is \( \cos 2 \theta \). We can rewrite this using the double angle formula for cosine: \( \cos 2\theta = 1 - 2 \sin^2 \theta \).

Step 2: Rewrite Right Side Using Secant Definition

The right side of the equation is \( \frac{2 - \sec^2 \theta}{\sec^2 \theta} \). Recall that \( \sec \theta = \frac{1}{\cos \theta} \), which means \( \sec^2 \theta = \frac{1}{\cos^2 \theta} \) or \( \cos^2 \theta = \frac{1}{\sec^2 \theta} \).

Step 3: Simplify the Right Side

Start by simplifying the expression \( \frac{2 - \sec^2 \theta}{\sec^2 \theta} \). This can be rewritten using a common denominator: \( 2 - \sec^2 \theta = \frac{2 \cos^2 \theta - 1}{\cos^2 \theta} \).

Step 4: Connect with Known Identities

Notice \( 2 \cos^2 \theta - 1 \) is another form of the double angle identity for cosine, so \( 2 \cos^2 \theta - 1 = \cos 2\theta \).

Step 5: Conclusion

Both sides simplify to \( \cos 2\theta \), thus proving that \( \cos 2 \theta = \frac{2 - \sec^2 \theta}{\sec^2 \theta} \) is an identity.

Key Concepts

Double Angle Formula

The Double Angle Formula is a useful trigonometric identity that allows us to express trigonometric functions of double angles in terms of single angles. For the cosine function, the double angle formula is:\[ \cos 2\theta = \cos^2 \theta - \sin^2 \theta \]There are equivalent forms of this identity that are often used, depending on the problem at hand:

• \( \cos 2\theta = 2\cos^2 \theta - 1 \)
• \( \cos 2\theta = 1 - 2\sin^2 \theta \)

These alternate forms come from the Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1 \). When using the double angle formula, you can choose the form that simplifies your problem most effectively. In proving identities, selecting the appropriate form can lead to quicker and simpler solutions.

Secant Function

The secant function is one of the six fundamental trigonometric functions. It's defined as the reciprocal of the cosine function:\[ \sec \theta = \frac{1}{\cos \theta} \]Given this relationship, the square of the secant function is:\[ \sec^2 \theta = \frac{1}{\cos^2 \theta} \]The secant function is particularly useful in identities and solving equations where the cosine function appears in the denominator. Because secant is related directly to cosine, it offers insights into relationships where division by cosine plays a role, as seen in the identity verification process. It's important to remember this connection, as it helps simplify expressions involving both cosine and secant.

Cosine Function

The cosine function is one of the fundamental trigonometric functions and is vital in representing rotational and wave phenomena. Defined on the unit circle, \( \cos \theta \) returns the x-coordinate of a point at an angle \( \theta \) from the positive x-axis. Its key properties include:

• It has a range from -1 to 1.
• It is periodic with a period of \( 2\pi \), meaning \( \cos(\theta + 2\pi) = \cos \theta \).
• It is even-symmetric, which means \( \cos(-\theta) = \cos \theta \).

In the context of the problem, understanding how cosine functions can be expressed in terms of secant and other forms through identities helps in verifying and simplifying trigonometric equations. Its interaction with other trigonometric functions often plays a central role in proofs and solutions in trigonometry.