Graph each expression and use the graph to make a conjecture, predicting what might be an identity. Then verify your conjecture algebraically.
csc x - cot x

Verified step by step guidance

1. Graph the expression \( y = \csc x - \cot x \) using a graphing calculator or software to observe its behavior.
2. Identify any patterns or symmetries in the graph that might suggest a trigonometric identity.
3. Consider known trigonometric identities involving \( \csc x \) and \( \cot x \), such as \( \csc x = \frac{1}{\sin x} \) and \( \cot x = \frac{\cos x}{\sin x} \).
4. Rewrite the expression \( \csc x - \cot x \) in terms of sine and cosine: \( \frac{1}{\sin x} - \frac{\cos x}{\sin x} \).
5. Simplify the expression: Combine the terms over a common denominator to verify if it simplifies to a known identity.

Key Concepts

Cosecant and Cotangent Functions

The cosecant function, csc(x), is the reciprocal of the sine function, defined as csc(x) = 1/sin(x). The cotangent function, cot(x), is the reciprocal of the tangent function, defined as cot(x) = cos(x)/sin(x). Understanding these functions is crucial for analyzing the expression csc(x) - cot(x) and their behavior on a graph.

Graphing Trigonometric Functions

Graphing trigonometric functions involves plotting their values over a specified interval, typically from 0 to 2π for periodic functions. Observing the graphs of csc(x) and cot(x) helps identify patterns, intersections, and potential identities. This visual representation is essential for making conjectures about relationships between trigonometric expressions.

Verifying Trigonometric Identities

Verifying trigonometric identities requires algebraic manipulation to show that two expressions are equivalent. This often involves using fundamental identities, such as Pythagorean identities or reciprocal identities, to transform one side of the equation into the other. This process is vital for confirming conjectures made from the graph of csc(x) - cot(x).