Problem 67 - Algebra and Trigonometry with Analytic Geometry

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 identity is verified: \( \sec ^{2} 3\theta \csc ^{2} 3\theta = \sec ^{2} 3\theta + \csc ^{2} 3\theta \).

Step 1: Understand the Trigonometric Identity

The goal is to simplify or transform \( \sec^2 3\theta \csc^2 3\theta \) and verify it is equal to \( \sec^2 3\theta + \csc^2 3\theta \). Recall \( \sec \theta = \frac{1}{\cos \theta} \) and \( \csc \theta = \frac{1}{\sin \theta} \). Also, remember the identity \( \sin^2 \theta + \cos^2 \theta = 1 \).

Step 2: Rewrite Secant and Cosecant

Express \( \sec^2 3\theta \) and \( \csc^2 3\theta \) in terms of sine and cosine: \( \sec^2 3\theta = \frac{1}{\cos^2 3\theta} \) and \( \csc^2 3\theta = \frac{1}{\sin^2 3\theta} \). Consequently, the left-hand side becomes \( \frac{1}{\cos^2 3\theta} \times \frac{1}{\sin^2 3\theta} \).

Step 3: Simplify the Complex Fraction

The expression \( \frac{1}{\cos^2 3\theta \sin^2 3\theta} \) is a complex fraction, which can be written as \( \frac{1}{\sin^2 3\theta \cos^2 3\theta} \).

Step 4: Use Trigonometric Identity to Simplify Further

Recognize that \( \sin^2 3\theta \cos^2 3\theta \) is part of the identity \( \sin^2 3\theta + \cos^2 3\theta = 1 \). However, for simplification purposes, consider reciprocal manipulation: \( \frac{1}{\sin^2 3\theta \cos^2 3\theta} = \frac{1}{\frac{1}{\sec^2 3\theta} + \frac{1}{\csc^2 3\theta}} \).

Step 5: Apply Reciprocal to Split Fraction

Apply the reciprocal in parts: \( \frac{1}{\sin^2 3\theta \cos^2 3\theta} \) becomes adding the reciprocals \( \frac{1}{\sec^2 3\theta} + \frac{1}{\csc^2 3\theta} \), seen as \( \sec^2 3\theta + \csc^2 3\theta \). This verifies the original statement.

Key Concepts

Secant Function

In trigonometry, the secant function is closely related to the cosine function. As a reminder, the secant of an angle \( \theta \) is defined as the reciprocal of the cosine of that angle. This means that:

• \( \sec \theta = \frac{1}{\cos \theta} \)
• \( \sec^2 \theta = \left(\frac{1}{\cos \theta}\right)^2 = \frac{1}{\cos^2 \theta} \)

Understanding the secant function is essential because it frequently appears in trigonometric identities and equations. It is important to keep in mind that secant has undefined values at angles where \( \cos \theta = 0 \), such as \( \theta = \frac{\pi}{2} \), \( \frac{3\pi}{2} \), etc. This is because dividing by zero in mathematics is undefined.
By using the definition of the secant, you can simplify expressions and solve trigonometric equations more effectively. Whenever you encounter a problem involving the secant function, remember its reciprocal relationship with cosine, and consider rewriting it in terms of cosine for easier manipulation.

Cosecant Function

Similar to the secant function, the cosecant function is the reciprocal of the sine function. Here's how it's defined:

• \( \csc \theta = \frac{1}{\sin \theta} \)
• \( \csc^2 \theta = \left(\frac{1}{\sin \theta}\right)^2 = \frac{1}{\sin^2 \theta} \)

The cosecant function provides a way to work with sine indirectly, especially in scenarios where equations become complex. Just as with secant, it's crucial to note the angles where the sine function is zero, such as \( \theta = 0 \), \( \pi \), \( 2\pi \), etc., because the cosecant becomes undefined.
Both the cosecant and secant functions are particularly useful because they allow transformation of expressions into equivalent forms that might be more suitable for manipulation or simplification. Whether working with expressions or solving equations, switching between these different trigonometric forms can provide fresh insights and solution strategies.

Reciprocal Identities

Reciprocal identities are the foundation for understanding trigonometric functions like the secant, cosecant, and cotangent. They are called reciprocal because they involve taking the reciprocal (or inverse) of a primary trigonometric function, such as sine, cosine, and tangent.
These identities help in rewriting complex trigonometric expressions into simpler forms, easing their manipulation greatly. For example:

• \( \sec \theta = \frac{1}{\cos \theta} \)
• \( \csc \theta = \frac{1}{\sin \theta} \)
• \( \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \)

Using reciprocal identities, you can convert trigonometric expressions into different forms, which might be simpler to manage depending on the problem at hand. By recognizing these identities, you can open up more pathways to verify equations, prove identities, or solve trigonometric equations effectively.
Incorporating reciprocal identities into your problem-solving toolbox will allow for more flexibility and efficiency, particularly with complex expressions like \( \sec^2 3\theta \csc^2 3\theta = \sec^2 3\theta + \csc^2 3\theta \). This identity highlights how understanding and applying these reciprocal relationships make complex-looking equations approachable and solvable.