Problem 3 - Precalculus with Calculus Previews

Question

Use the fundamental identities and the even-odd identities to simplify each expression. $$ \frac{\sin \theta}{\csc \theta}+\frac{\cos \theta}{\sec \theta} $$

Step-by-Step Solution

Answer

The simplified expression is 1.

Step 1: Replace Cosecant and Secant

The expression is \( \frac{\sin \theta}{\csc \theta} + \frac{\cos \theta}{\sec \theta} \). First, replace \( \csc \theta \) and \( \sec \theta \) with their fundamental identities: \( \csc \theta = \frac{1}{\sin \theta} \) and \( \sec \theta = \frac{1}{\cos \theta} \). Substitute these into the expression.

Step 2: Simplify Fractions

Substitute the identities: \( \frac{\sin \theta}{\frac{1}{\sin \theta}} + \frac{\cos \theta}{\frac{1}{\cos \theta}} \). Simplify each fraction: \( \frac{\sin \theta}{\frac{1}{\sin \theta}} = \sin \theta \times \sin \theta = \sin^2 \theta \) and \( \frac{\cos \theta}{\frac{1}{\cos \theta}} = \cos \theta \times \cos \theta = \cos^2 \theta \). So the expression becomes \( \sin^2 \theta + \cos^2 \theta \).

Step 3: Apply Pythagorean Identity

Recall the Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1 \). Use this identity to simplify \( \sin^2 \theta + \cos^2 \theta \) to \( 1 \).

Key Concepts

Fundamental Identities

In trigonometry, fundamental identities are the building blocks used to simplify and solve various trigonometric expressions. They establish relationships between trigonometric functions that are always true. Some of the most vital fundamental identities include:

• Reciprocal identities: These express basic trigonometric functions in terms of their reciprocals, such as \( \csc \theta = \frac{1}{\sin \theta} \) and \( \sec \theta = \frac{1}{\cos \theta} \).
• Tangent and cotangent identities: For instance, \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) and \( \cot \theta = \frac{\cos \theta}{\sin \theta} \).

These identities allow for the transformation of the given expression. In our exercise, the functions cosecant and secant were expressed in terms of sine and cosine using reciprocal identities, respectively. This simplification forms the foundation of manipulating trigonometric expressions efficiently, making complex calculations more manageable.

Even-Odd Identities

Another essential aspect of trigonometry is understanding even-odd identities. These identities define the behavior of trigonometric functions under reflection across the y-axis.

• Even functions: These are symmetric about the y-axis. For example, \( \cos(-\theta) = \cos \theta \) and \( \sec(-\theta) = \sec \theta \).
• Odd functions: These exhibit symmetry about the origin. For instance, \( \sin(-\theta) = -\sin \theta \) and \( \tan(-\theta) = -\tan \theta \).

Even-odd identities are crucial when simplifying expressions, especially when dealing with transformations. By knowing whether a function retains its value or changes its sign, you can predict and simplify the behavior of expressions more effectively. In the context of our exercise, although directly not used, understanding these identities provides a deeper insight as to how functions behave when varied by negative inputs.

Pythagorean Identity

The Pythagorean identity lies at the heart of numerous trigonometric simplifications, embodying the essence of the fundamental relationships between sine and cosine. The most well-known Pythagorean identity is:\[ \sin^2 \theta + \cos^2 \theta = 1 \]This identity springs from the Pythagorean theorem applied to a right triangle, representing the sum of the squares of sine and cosine equalling one. This powerful identity played a key role in simplifying our expression in the provided exercise. Once the initial expression was transformed using fundamental identities, recognizing and applying the Pythagorean identity allowed for its final reduction to a basic value of 1. Besides the primary identity, other forms exist, re-expressed to involve tangent or cosecant, which are:

• \( 1 + \tan^2 \theta = \sec^2 \theta \)
• \( 1 + \cot^2 \theta = \csc^2 \theta \)

These variations further emphasize the interconnectedness of trigonometric functions and their properties, serving as a toolbox for simplifying expressions effectively.