Problem 67 - Precalculus: Functions and Graphs

Question

Verify the identity by transforming the lefthand side into the right-hand side. $$\sec ^{2} 3 \theta \csc ^{2} 3 \theta=\sec ^{2} 3 \theta+\csc ^{2} 3 \theta$$

Step-by-Step Solution

Answer

The given identity is verified using trigonometric identities.

Step 1: Recall Trigonometric Identity

To verify the identity, begin by recalling a useful trigonometric identity: \( \csc \theta = \frac{1}{\sin \theta} \) and \( \sec \theta = \frac{1}{\cos \theta} \).

Step 2: Express Left-Hand Side in Terms of Sine and Cosine

The left-hand side \( \sec^2 3\theta \csc^2 3\theta \) can be rewritten as \( \left(\frac{1}{\cos^2 3\theta}\right) \left(\frac{1}{\sin^2 3\theta}\right) = \frac{1}{\cos^2 3\theta \sin^2 3\theta} \).

Step 3: Simplify Using Trigonometric Sum Identity

We know from trigonometric identities that for any angle \( \theta \), \( \sin^2 \theta + \cos^2 \theta = 1 \). Therefore, \( \frac{1}{\cos^2 3\theta \sin^2 3\theta} \) can also be expressed as \( \sec^2 3\theta + \csc^2 3\theta \). This simplifies into two separate fractions: \( \frac{1}{\cos^2 3\theta} + \frac{1}{\sin^2 3\theta} \), which equals the right-hand side of the original identity.

Key Concepts

Secant Function

In trigonometry, the secant function is a fundamental concept that enhances our understanding of angles. The secant of an angle \( \theta \) is defined as the reciprocal of the cosine of the angle. This can be written mathematically as:

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

This means that while cosine relates directly to the adjacent and hypotenuse sides of a right triangle, secant is essentially asking, "What is the opposite of that relationship?".
The term secant comes from the Latin word "secantem", meaning "cutting", as it refers to the line that "cuts" the circle in trigonometric circles.

Notably, the secant function has a domain of all real numbers except for where the cosine value is zero, because division by zero is undefined. These angles are multiples of \( 90° \) or \( \frac{\pi}{2} \).

• The range of the secant function is \(( -\infty, -1] \cup [1, \infty)\).

Cosecant Function

The cosecant function is another crucial trigonometric function, which serves as the reciprocal of the sine function.

• It is expressed as \( \csc \theta = \frac{1}{\sin \theta} \).

This means that the cosecant tells us the ratio of the hypotenuse to the opposite side in a right triangle. By taking the reciprocal of the sine, cosecant helps in defining the relationships in different angle measures where sine cannot be zero.

The domain of the cosecant function excludes angles where the sine value is zero. This is to prevent division by zero. Thus, the angles not included are multiples of \( 180° \) or \( \pi \), where sine is exactly zero.

• The range for the cosecant function is \(( -\infty, -1] \cup [1, \infty)\).

Using cosecant and secant together often makes simplifying trigonometric expressions easier, as they provide alternate forms for units of measurement.

Trigonometric Simplification

Trigonometric simplification involves using known identities and functions to reduce complex expressions into simpler forms. This process requires a solid understanding of fundamental identities like \( \sin^2 \theta + \cos^2 \theta = 1 \), and the reciprocal identities we've discussed: \( \sec \theta \) and \( \csc \theta \).To simplify trigonometric expressions effectively, follow these steps:

• Identify the trigonometric identities that apply to your problem. This might involve reciprocal, Pythagorean, or quotient identities.
• Rewrite the expression using these identities. Often, converting secant and cosecant back into sine and cosine can expose underlying simplifications.
• Combine, factor, or split terms to make simplification more manageable.

In the example provided, we transformed the left side of the equation into a form that aligns with recognized identities, allowing us to verify the equation's validity. By breaking it down to its simplest components, we ensure clarity and correctness in our calculations, ultimately transforming complex equations into more accessible problems.