Problem 39 - Precalculus: Functions and Graphs

Question

Verify the Identity. $$\sec \theta+\csc \theta-\cos \theta-\sin \theta=\sin \theta \tan \theta+\cos \theta \cot \theta$$

Step-by-Step Solution

Answer

Both sides simplify to \( \frac{\sin \theta + \cos \theta - \cos \theta \sin \theta (\cos \theta + \sin \theta)}{\cos \theta \sin \theta} \), confirming the identity.

Step 1: Simplify the Left Side

Start by simplifying the left side of the identity: \( \sec \theta + \csc \theta - \cos \theta - \sin \theta \). Express \( \sec \theta \) and \( \csc \theta \) in terms of sine and cosine: \( \frac{1}{\cos \theta} + \frac{1}{\sin \theta} - \cos \theta - \sin \theta \). Combine these into a single fraction: \( \frac{\sin \theta + \cos \theta - \cos \theta \sin \theta (\cos \theta + \sin \theta)}{\cos \theta \sin \theta} \).

Step 2: Simplify the Right Side

Now work on the right side of the identity: \( \sin \theta \tan \theta + \cos \theta \cot \theta \). Substitute the trigonometric identities for \( \tan \theta \) and \( \cot \theta \): \( \sin \theta \times \frac{\sin \theta}{\cos \theta} + \cos \theta \times \frac{\cos \theta}{\sin \theta} \). This gives: \( \frac{\sin^2 \theta}{\cos \theta} + \frac{\cos^2 \theta}{\sin \theta} \).

Step 3: Combine Terms and Simplify

To verify the identity, notice that both sides have a common denominator of \( \cos \theta \sin \theta \). Rewriting the right side: \( \frac{\sin^3 \theta + \cos^3 \theta}{\cos \theta \sin \theta} \). Recognize that this is equal to \( \sin \theta + \cos \theta - \cos \theta \sin \theta (\cos \theta + \sin \theta) \) from the left side.

Step 4: Final Verification

After organizing and simplifying both sides, they equate to \( \frac{\sin \theta + \cos \theta - \cos \theta \sin \theta (\cos \theta + \sin \theta)}{\cos \theta \sin \theta} \), confirming that the original equation is an identity.

Key Concepts

Understanding Secant and Cosecant

Secant and cosecant are reciprocal trigonometric functions. While they might seem intimidating at first, once you understand their basics, they become much simpler. \( \sec \theta \) (secant) is the reciprocal of \( \cos \theta \). This means that \( \sec \theta = \frac{1}{\cos \theta} \). Similarly, \( \csc \theta \) (cosecant) is the reciprocal of \( \sin \theta \), so \( \csc \theta = \frac{1}{\sin \theta} \). Understanding these can greatly help in simplifying expressions, especially when verifying trigonometric identities.

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

In our exercise, we replaced \( \sec \theta \) and \( \csc \theta \) with their equivalents in terms of sine and cosine to simplify the given expression. Knowing how to do this switch is crucial for managing more complex trigonometric problems.

The Basics of Sine and Cosine

Sine and cosine are fundamental to trigonometry and are often the building blocks for more complex functions like secant and cosecant. They are based on right-angled triangles on a unit circle.

• \( \sin \theta \) is the ratio of the opposite side to the hypotenuse.
• \( \cos \theta \) is the ratio of the adjacent side to the hypotenuse.

In calculus and many trigonometric problems, you will frequently express trigonometric identities in terms of sine and cosine. For instance, classic identities like \( \sin^2 \theta + \cos^2 \theta = 1 \) emerge from these relationships. When tackling the exercise, expressing secant and cosecant in terms of sine and cosine allowed us to manipulate and simplify the identity more effectively.

Simplifying Expressions in Trigonometry

Simplifying expressions can make complex trigonometric identities manageable. The process typically involves rewriting terms using basic trigonometric identities or converting them into a form that reveals a cancellation or common factor.

• Look for opportunities to replace reciprocal functions with their definitions, like \( \sec \theta \) and \( \csc \theta \).
• Utilize fundamental identities, such as \( \sin^2 \theta + \cos^2 \theta = 1 \), to simplify.
• Combine fractions by finding a common denominator, which simplifies the equation.

In our original exercise, simplifying each side of the equation helped us to find a common expression and verify the identity. By converting complicated terms into fractions and combining them properly, both sides of the equation could be shown to have the same value. This methodical approach is often essential, especially for verifying identities in trigonometry.