Problem 58 - A Graphical Approach to Precalculus with Limits

Question

Write expression in terms of sine and cosine, and simplify it. (The final expression does not have to be in terms of sine and cosine.) $$(\sec \theta+\csc \theta)(\cos \theta-\sin \theta)$$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{\cos^2 \theta - \sin^2 \theta}{\cos \theta \sin \theta} \).

Step 1: Rewrite Secant and Cosecant

Recall the identities \( \sec \theta = \frac{1}{\cos \theta} \) and \( \csc \theta = \frac{1}{\sin \theta} \). Substitute these into the given expression: \( (\frac{1}{\cos \theta} + \frac{1}{\sin \theta})(\cos \theta - \sin \theta) \).

Step 2: Combine Fractions

Combine the fractions in the expression \( \frac{1}{\cos \theta} + \frac{1}{\sin \theta} \) by finding a common denominator, which is \( \cos \theta \sin \theta \). This gives: \( \frac{\sin \theta + \cos \theta}{\cos \theta \sin \theta} \).

Step 3: Multiply Expressions

Multiply the simplified fraction by the binomial in the original expression: \( \left(\frac{\sin \theta + \cos \theta}{\cos \theta \sin \theta} \right)(\cos \theta - \sin \theta) \).

Step 4: Distribute Terms

Distribute \( \cos \theta - \sin \theta \) across \( \sin \theta + \cos \theta \): \[ \frac{\sin \theta \cos \theta - \sin^2 \theta + \cos^2 \theta \cos \theta - \cos \theta \sin \theta}{\cos \theta \sin \theta} \].

Step 5: Simplify the Numerator

Return the numerator to a simplified state: \( \cos^2 \theta - \sin^2 \theta \). This uses the identities \( \sin \theta \cos \theta - \cos \theta \sin \theta = 0 \), removing the terms that cancel out.

Step 6: Write as Single Fraction

Express the remaining fraction as a single fraction: \[ \frac{\cos^2 \theta - \sin^2 \theta}{\cos \theta \sin \theta} \].

Step 7: Apply Pythagorean Identity

Using the identities \( \cos^2 \theta = 1 - \sin^2 \theta \) and \( \sin^2 \theta = 1 - \cos^2 \theta \), recognize that the numerator can be simplified further as \( (1 - \sin^2 \theta) - (1 - \cos^2 \theta) = \cos^2 \theta - \sin^2 \theta \).

Key Concepts

Sine and Cosine

In trigonometry, the sine and cosine functions are fundamental. They are commonly used in various mathematical expressions and apply to right triangles and unit circles.
- **Sine (\( \sin \theta \)):** This function represents the ratio of the length of the side opposite the angle \( \theta \) to the hypotenuse of a right triangle.
- **Cosine (\( \cos \theta \)):** This function is the ratio of the length of the adjacent side to the hypotenuse.
Understanding sine and cosine is crucial for simplifying trigonometric expressions.
In the given exercise, the task involves expressing quantities in terms of \( \sin \theta \) and \( \cos \theta \). By substituting \( \sec \theta = \frac{1}{\cos \theta} \) and \( \csc \theta = \frac{1}{\sin \theta} \) into the original expression, \( (\sec \theta+\csc \theta)(\cos \theta-\sin \theta) \), the expression is translated entirely into sine and cosine terms. This transformation is essential for simplifying and solving complex trigonometric identities.

Secant and Cosecant

Secant and cosecant are less frequently encountered than sine and cosine. These functions are reciprocals of cosine and sine, respectively.
- **Secant (\( \sec \theta \)):** This is the reciprocal of cosine, \( \sec \theta = \frac{1}{\cos \theta} \).
- **Cosecant (\( \csc \theta \)):** This is the reciprocal of sine, \( \csc \theta = \frac{1}{\sin \theta} \).
In the exercise, the expression \( \sec \theta + \csc \theta \) is rewritten using these identities. The reciprocal nature of secant and cosecant means that they can be expressed in terms of sine and cosine. Simplifying trigonometric expressions often involves substituting secant and cosecant with their respective reciprocal forms, reducing complexity and facilitating further manipulation.

Pythagorean Identity

The Pythagorean Identity is a fundamental relationship in trigonometry involving the sine and cosine functions. It expresses the intrinsic geometric properties of these functions:
- The basic Pythagorean Identity is: \( \sin^2 \theta + \cos^2 \theta = 1 \).
This identity helps simplify expressions by resolving terms involving squares of sine and cosine. During the solution of the exercise, it was used to simplify the numerator from \( \cos^2 \theta - \sin^2 \theta \) to a more manageable form. By applying the identities \( \cos^2 \theta = 1 - \sin^2 \theta \) and \( \sin^2 \theta = 1 - \cos^2 \theta \), one can derive other meaningful relationships. The Pythagorean Identity ensures that complex expressions reduce into simpler, more usable forms, clearing the path to arriving at the final simplified answer.