Problem 23 - Precalculus: Functions and Graphs

Question

Verify the identity. \(\sec 2 \theta=\frac{\sec ^{2} \theta}{2-\sec ^{2} \theta}\)

Step-by-Step Solution

Answer

The given identity is verified as true.

Step 1: Recall the Identity for Double Angle

To start verifying the identity, we need to recall that \( \sec 2\theta = \frac{1}{\cos 2\theta} \). We will compare this with the right side equation to prove the identities are equivalent.

Step 2: Express the Right Side in Terms of Cosine

The right side of the given identity is \( \frac{\sec^2 \theta}{2 - \sec^2 \theta} \). Rewriting \( \sec^2 \theta \) as \( \frac{1}{\cos^2 \theta} \), the expression becomes \( \frac{\frac{1}{\cos^2 \theta}}{2 - \frac{1}{\cos^2 \theta}} \).

Step 3: Combine the Fractions

Rewrite the denominator of the right side as a single fraction: \(2 - \frac{1}{\cos^2 \theta} = \frac{2\cos^2 \theta - 1}{\cos^2 \theta} \). Thus, the right side becomes \( \frac{\frac{1}{\cos^2 \theta}}{\frac{2\cos^2 \theta - 1}{\cos^2 \theta}} \).

Step 4: Simplify the Fraction

By simplifying the complex fraction, \( \frac{\frac{1}{\cos^2 \theta}}{\frac{2\cos^2 \theta - 1}{\cos^2 \theta}} \) becomes \( \frac{1}{2\cos^2 \theta - 1} \).

Step 5: Verify the Identity by Using Cosine Double Angle Formula

Recall the double angle formula for cosine: \( \cos 2\theta = 2\cos^2 \theta - 1 \). Thus, \( \frac{1}{\cos 2\theta} = \frac{1}{2\cos^2 \theta -1} \). This shows the identity \( \sec 2\theta = \frac{\sec^2 \theta}{2-\sec^2 \theta} \) is true.

Key Concepts

Double Angle Formulas

Double angle formulas are key tools in trigonometry. They help in simplifying expressions involving trigonometric functions when the angle is doubled. Specifically, for cosine, the double angle formula is:\[\cos 2\theta = 2\cos^2 \theta - 1\]This formula expresses the cosine of double the angle in terms of the square of the cosine of the original angle. It's very handy when dealing with complex trigonometric identities.

These formulas are not just restricted to the cosine function. There are also double angle formulas for sine and tangent:

• \(\sin 2\theta = 2\sin \theta \cos \theta\)
• \(\tan 2\theta = \frac{2\tan \theta}{1 - \tan^2 \theta}\)

Understanding these formulas helps you manipulate and prove identities like the one in the exercise by expressing various trigonometric functions in new forms.

In the case of the given problem, we use the cosine double angle formula to simplify the expression for secant so that it matches the given identity.

Secant Function

The secant function is one of the lesser-known but equally important trigonometric functions. It is defined as the reciprocal of the cosine function:\[\sec \theta = \frac{1}{\cos \theta}\]It's particularly useful when dealing with reciprocal identities and simplifying expressions where the cosine function appears in the denominator.

In trigonometric identities, like in the given problem, knowing how to transform secant into terms of cosine is crucial. For \(\sec 2\theta\), we use the relationship:\[\sec 2\theta = \frac{1}{\cos 2\theta}\]Having this understanding allows you to rewrite expressions in terms of cosine, which can simplify the process of verifying complex identities.

When working with identities, it's often necessary to express secant as \(\sec^2 \theta\), leading to additional simplifications and transformations.

Cosine Function

The cosine function is foundational in trigonometry, and its properties are frequently used in verifying and proving identities. Cosine, related to the adjacent and hypotenuse sides of a right triangle, is defined as:\[\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}\]In the unit circle context, it is the horizontal distance from the origin to the point on the circle.

In trigonometric identities, the cosine function often appears because its transformations and properties can relate different angles and simplify expressions. In our exercise, applying the double angle formula for cosine was pivotal in transforming the identity into a verifiable form.

Remembering these key formulas and their manipulations becomes significantly beneficial:

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

Through practice, using these relations becomes intuitive, making the verification or transformation of identities like the one presented swift and accurate.