Problem 24 - Precalculus with Limits

Question

In Exercises 9-50, verify the identity \( \dfrac{\sec^2 \theta - 1}{1 - \cos \theta} = \sec \theta \)

Step-by-Step Solution

Answer

After using these steps to manipulate the left-hand side of the equation, it was successfully shown that it equals the right hand side \( \sec \theta \). Therefore, the identity has been verified.

Step 1: Express Secant in terms of Cosine

Rewrite \( \sec^2 \theta \) as \( \dfrac{1}{\cos^2 \theta} \). This gives us \( \dfrac{\frac{1}{\cos^2 \theta} - 1}{1 - \cos \theta} \).

Step 2: Simplify the Equation

Now, we can find a common denominator to combine the terms under the numerator and get \( \frac{1 - \cos^2 \theta}{\cos^2 \theta} \) in the numerator. This simplifies the equation to \( \frac{\frac{1 - \cos^2 \theta}{\cos^2 \theta}}{1 - \cos \theta} \).

Step 3: Use Pythagorean Identity

We use the Pythagorean Identity \(sin^2 \theta = 1 - \cos^2 \theta \). This step simplifies the numerator to \( \frac{sin^2 \theta}{\cos^2 \theta} \). Now, the expression is \( \frac{\frac{sin^2 \theta}{\cos^2 \theta}}{1 - \cos \theta} \).

Step 4: Cancel out the Common term

Now cancel out the common factor \( (1 - cos \theta) \) from the numerator and denominator which gives us \( \frac{sin \theta}{cos \theta} \).

Step 5: Express the equation in terms of Secant

Remembering that \( \sec \theta = \frac{1}{cos \theta} \) and \( \frac{sin \theta}{cos \theta} = tan \theta \). This simplifies to \( \sec \theta \), which is the right side of the identity, where the \( \sec \theta \) is the reciprocal of \( cos \theta \).

Key Concepts

Pythagorean Identity

Understanding the Pythagorean Identity is crucial when it comes to verifying trigonometric identities. It is an equation expressing a fundamental relation between the sine and cosine functions. The identity can be written as \(sin^2 \theta + cos^2 \theta = 1\).

In practice, by rearranging this equation, you can isolate either the sine or cosine squared term, which allows us to transform an expression involving \(sin^2 \theta\) or \(cos^2 \theta\) into the other corresponding term. This is particularly useful when trigonometric functions in an identity appear complex or not directly comparable, as we simplify the expressions to forms that are easier to manipulate.

Secant Function

The secant function, denoted as \(sec \theta\), is one of the six fundamental trigonometric functions. It is the reciprocal of the cosine function, meaning \(sec \theta = \frac{1}{cos \theta}\). It is important to recognize secant as the flip-side of cosine when dealing with trigonometric identities. In terms of triangles, if you have a right triangle with an adjacent side (next to the angle \( \theta\)) and a hypotenuse (the triangle's longest side), the secant of the angle is the ratio of the length of the hypotenuse to the length of the adjacent side.

The secant function often appears in more complex trigonometric expressions and understanding its relationship with the cosine function aids greatly in simplifying and verifying identities.

Cosine Function

The cosine function is a primary trigonometric function that relates the angle in a right triangle to the ratio of the adjacent side to the hypotenuse. In formulaic terms, \(cos \theta \ = \frac{adjacent}{hypotenuse}\). It is an even function, meaning that \(cos(\theta) = cos(-\theta)\), and it has a range of [-1, 1].

In verifying trigonometric identities, the cosine function is often manipulated algebraically or transformed into another related function (like the secant function) to facilitate the proof. Being comfortable with these transformations and relationships between trigonometric functions is essential.

Sine Function

The sine function, represented by \(sin \theta\), is another fundamental trigonometric ratio. For an angle \( \theta\) in a right-angled triangle, it is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. In simple terms, \(sin \theta \ = \frac{opposite}{hypotenuse}\).

Just like the cosine function, the sine function plays a significant role in verifying trigonometric identities. Coupled with the Pythagorean Identity, the sine function allows us to interchange expressions involving \(sin^2 \theta\) and \(cos^2 \theta\), thereby simplifying the verification process for many trigonometric identities.