Problem 112 - Precalculus with Limits

Question

In Exercises 111 - 124, verify the identity. \( \sec 2 \theta = \dfrac{\sec^2 \theta}{2 - \sec^2 \theta} \)

Step-by-Step Solution

Answer

Using the Pythagorean theorem and the double angle formula for secants, we were able to successfully verify the identity.

Step 1: Express everything in terms of \(\tan \theta \)

Rewrite the identity replacing \( \sec^2 \theta \) with \(1 + \tan^2 \theta \) This turns the equation into \( \sec 2 \theta = \dfrac{1 + \tan^2 \theta}{2 - 1 - \tan^2 \theta} \)

Step 2: Simplify the equation

Now simplify the equation formed in the previous step. The new equation is \( \sec 2 \theta = \dfrac{1 + \tan^2 \theta}{1 - \tan^2 \theta} \)

Step 3: Use the double angle formula for secant

Now use the double angle formula \( \sec 2 \theta = 1/ \cos 2\theta \) and the relation \( \cos 2 \theta = 1 - 2 \tan^2 \theta / (1 + \tan^2 \theta) \). This gives \( \sec 2 \theta = \dfrac{1 + \tan^2 \theta}{1 - \tan^2 \theta} \)

Step 4: Confirm both sides are equal

Since both sides of the equation are equal, the identity has been verified.

Key Concepts

Secant Function

The secant function, often denoted as \( \sec \theta \), is one of the six fundamental trigonometric functions. It is the reciprocal of the cosine function. That means we can express it as:

• \( \sec \theta = \frac{1}{\cos \theta} \)

The secant function is helpful in solving various trigonometric identities and equations, especially when angles are involved. It relates directly to the circle's radius over its x-coordinate when a circle is centered at the origin. Additionally, in many problems, we use \( \sec^2 \theta = 1 + \tan^2 \theta \) to simplify expressions. These relationships are critical because they form part of foundational trigonometric identities used for verification or simplification tasks.

Double Angle Formulas

Double angle formulas are trigonometric identities that predict the value of trigonometric functions of double angles. For instance, the secant double angle formula is derived from the double angle formula of cosine. The formula used is:

• \( \cos 2 \theta = 1 - 2 \tan^2 \theta / (1 + \tan^2 \theta) \)

This relationship allows us to transform trigonometric expressions involving double angles into more manageable expressions. It replaces the need for calculating specific trigonometric functions at multiplied angles, making identity verification easier. The double angle formula is not just limited to cosine but extends to sine and tangent, providing a toolkit for solving diverse identity problems. For secant, use the relation \( \sec 2\theta = \frac{1}{\cos 2\theta} \). This is an essential step when verifying complex trigonometric identities.

Tangent

The tangent of an angle, represented by \( \tan \theta \), is another primary trigonometric function. It's defined by the ratio of the sine to the cosine of an angle:

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

The tangent function describes the slope of the line formed from the origin of a unit circle to the point on the circle. It is periodic and has important relationships with other trig functions, such as the Pythagorean identity \( 1 + \tan^2 \theta = \sec^2 \theta \). These relations are often used to simplify expressions, like the one in the given identity. In trigonometry, understanding how tangent interacts with other functions helps to transform and verify equations. Here, converting \( \sec^2 \theta \) to \( 1 + \tan^2 \theta \) enables easier manipulation and simplification of identities.

Verifying Identities

Verifying trigonometric identities involves demonstrating that two sides of an equation are equivalent. It requires breaking down expressions into known identities or simpler functions until both sides match. This process becomes a logical proof where you use known relationships like:

• Pythagorean Identities (e.g., \( 1 + \tan^2 \theta = \sec^2 \theta \))
• Reciprocal Identities (e.g., \( \sec \theta = \frac{1}{\cos \theta} \))
• Double Angle Formulas

When verifying, start by transforming complex expressions, applying identities step-by-step, and ensuring each transformation retains equivalency. In cases like \( \sec 2 \theta = \frac{\sec^2 \theta}{2 - \sec^2 \theta} \), understanding and manipulating these identities were critical. By expressing terms via \( \tan \theta \) and employing double angle relations, you can systematically prove an identity's correctness. The key is consistent application of identities and simplifying each side of the equation creatively.