Problem 62 - A Graphical Approach to Precalculus with Limits

Question

Perform indicated operation and simplify the result. $$\frac{\sec x}{\csc x}+\frac{\csc x}{\sec x}$$

Step-by-Step Solution

Answer

The simplified form is \( \sin x \cos x (2 + \tan^2 x + \cot^2 x) \).

Step 1: Identify the Terms and Common Denominators

The given expression is \( \frac{\sec x}{\csc x} + \frac{\csc x}{\sec x} \). To simplify, first notice that each part of the expression has different denominators. A common denominator will help to combine these fractions. Identify that the common denominator of \( \csc x \) and \( \sec x \) is \( \csc x \sec x \).

Step 2: Rewrite Each Term with the Common Denominator

Rewriting each fraction using the common denominator \( \csc x \sec x \), we get \( \frac{\sec^2 x}{\csc x \sec x} \) for the first term and \( \frac{\csc^2 x}{\csc x \sec x} \) for the second term. Now the expression is: \( \frac{\sec^2 x + \csc^2 x}{\csc x \sec x} \).

Step 3: Simplify Using Trigonometric Identities

Recall the identities \( \sec^2 x = 1 + \tan^2 x \) and \( \csc^2 x = 1 + \cot^2 x \). Plug these into the expression to transform it: \[ \frac{(1 + \tan^2 x) + (1 + \cot^2 x)}{\csc x \sec x} = \frac{2 + \tan^2 x + \cot^2 x}{\csc x \sec x} \].

Step 4: Evaluate Using Basic Trigonometric Simplification

Notice that \( \tan^2 x = \frac{\sin^2 x}{\cos^2 x} \) and \( \cot^2 x = \frac{\cos^2 x}{\sin^2 x} \). Simplifying \( \tan^2 x + \cot^2 x \) yields \( \frac{\sin^2 x}{\cos^2 x} + \frac{\cos^2 x}{\sin^2 x} = \frac{\sin^4 x + \cos^4 x}{\sin^2 x \cos^2 x} \).

Step 5: Simplify the Entire Expression

Combine simplifications to get the final form: \[ \frac{2 + \frac{\sin^4 x + \cos^4 x}{\sin^2 x \cos^2 x}}{\csc x \sec x} = \frac{2\sin^2 x \cos^2 x + \sin^4 x + \cos^4 x}{\sin^2 x \cos^2 x \csc x \sec x} \]. Further simplify by factoring and canceling common terms to reach the simplest form.

Key Concepts

Trigonometric Simplification

Understanding trigonometric simplification can greatly aid in solving various mathematical problems. It involves reducing complex trigonometric expressions into simpler forms, often by applying basic identities. In the given exercise, we start with the expression \( \frac{\sec x}{\csc x} + \frac{\csc x}{\sec x} \) and aim to make it less complex.

Trigonometric identities, such as those for secant and cosecant, are central to simplification:

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

By converting these into sine and cosine, the expression reduces to basic arithmetic operations. Further simplification often involves recognizing reciprocal identities and Pythagorean identities, which reveal underlying patterns and help cancel terms. Simplifying trigonometric expressions is powerful because it turns complex calculations into manageable steps, making it easier to work with them in equations.

Fraction Operations

Fraction operations are key when handling trigonometric expressions. The exercise demonstrates combining fractions, where each has different denominators. To add or subtract fractions, first identify a common denominator, which in this case is \( \csc x \sec x \).

Once the common denominator is identified, rewrite each fraction with it:

• The first term becomes \( \frac{\sec^2 x}{\csc x \sec x} \).
• The second term becomes \( \frac{\csc^2 x}{\csc x \sec x} \).

After rewriting, the expression can be combined as \( \frac{\sec^2 x + \csc^2 x}{\csc x \sec x} \). This not only eases computation but also sets the stage for further simplification using identities. Understanding these operations empowers students to tackle more complex expressions with confidence.

Trigonometric Functions

In trigonometry, functions like secant and cosecant support the exploration and simplification of expressions. Secant (\( \sec x \)) and cosecant (\( \csc x \)) are the reciprocals of cosine and sine, respectively. By expressing them in terms of sine and cosine, calculations become more straightforward.

Important identities apply:

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

These identities link secant and cosecant to tangent and cotangent, offering alternate pathways for expression manipulation. When simplifying, the identities unravel into expressions with familiar sine and cosine functions, often simplifying computations and enabling further reduction. A firm grasp of these functions and their identities aids the application of their properties in diverse trigonometric contexts.