For each expression in Column I, choose the expression from Column II that completes an identity.
2. csc x = ____


II
A. sin ^2 x/cos ^2 x
B.1/(sec ^2 x)
C. sin (-x)
D. csc ^2 x-cot ^2 x + sin ^2 x
E. tan x

Verified step by step guidance

1. Recall the definition of the cosecant function in terms of sine: \(\csc x = \frac{1}{\sin x}\).
2. Identify the expression in Column II that matches \(\frac{1}{\sin x}\) or is equivalent to it.
3. Verify that the chosen expression satisfies the identity by considering the domain where \(\sin x \neq 0\) to avoid division by zero.
4. Understand that this identity is fundamental because cosecant is the reciprocal of sine, which is a key concept in trigonometry.
5. Confirm that the expression from Column II correctly completes the identity \(\csc x = \frac{1}{\sin x}\).

Key Concepts

Reciprocal Trigonometric Functions

The cosecant function (csc x) is the reciprocal of the sine function. This means csc x = 1/sin x, which is fundamental for rewriting or completing trigonometric identities involving csc x.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. Recognizing and applying these identities helps simplify expressions and solve equations.

Domain and Range of Trigonometric Functions

Understanding the domain and range of functions like sine and cosecant is important because csc x is undefined where sin x = 0. This knowledge ensures correct application of identities and avoids division by zero.