Perform each indicated operation and simplify the result so that there are no quotients.
sec x/csc x + csc x/sec x

Verified step by step guidance

1. Identify the trigonometric identities: \( \sec x = \frac{1}{\cos x} \) and \( \csc x = \frac{1}{\sin x} \).
2. Rewrite the expression \( \frac{\sec x}{\csc x} + \frac{\csc x}{\sec x} \) using the identities: \( \frac{\frac{1}{\cos x}}{\frac{1}{\sin x}} + \frac{\frac{1}{\sin x}}{\frac{1}{\cos x}} \).
3. Simplify each fraction by multiplying the numerator and the denominator by the reciprocal: \( \frac{\sin x}{\cos x} + \frac{\cos x}{\sin x} \).
4. Recognize that \( \frac{\sin x}{\cos x} = \tan x \) and \( \frac{\cos x}{\sin x} = \cot x \).
5. Combine the expressions: \( \tan x + \cot x \).

Key Concepts

Secant and Cosecant Functions

Secant (sec) and cosecant (csc) are trigonometric functions defined as the reciprocals of cosine and sine, respectively. Specifically, sec x = 1/cos x and csc x = 1/sin x. Understanding these functions is crucial for manipulating expressions involving them, as they often appear in various trigonometric identities and equations.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions involves combining and reducing terms to achieve a more manageable form. This often includes using identities, such as the Pythagorean identities or reciprocal identities, to rewrite functions in terms of sine and cosine, which can help eliminate quotients and facilitate further calculations.

Common Denominators

When adding or subtracting fractions, finding a common denominator is essential. In the context of trigonometric functions, this means expressing each term with a shared base, allowing for straightforward addition or subtraction. This concept is particularly important in the given expression, as it enables the combination of sec x/csc x and csc x/sec x into a single simplified form.