For each expression in Column I, choose the expression from Column II that completes an identity.
2. csc 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 reciprocal identity that relates cosecant and sine: \(\csc x\) is the reciprocal of \(\sin x\).
3. Look for the expression in Column II that matches \(\frac{1}{\sin x}\) or an equivalent form.
4. Verify that substituting the chosen expression into the identity maintains equality for all valid values of \(x\) where \(\sin x \neq 0\).
5. Conclude that the correct expression completing the identity for \(\csc x\) is \(\frac{1}{\sin x}\).

Key Concepts

Reciprocal Trigonometric Identities

Reciprocal identities relate trigonometric functions to each other by expressing one as the reciprocal of another. For example, cosecant (csc x) is the reciprocal of sine (sin x), meaning csc x = 1/sin x. Understanding these identities helps simplify and transform trigonometric expressions.

Definition of Cosecant Function

The cosecant function, csc x, is defined as the ratio of the hypotenuse to the opposite side in a right triangle, or equivalently, as 1 divided by sin x. This definition is fundamental when working with trigonometric identities and solving equations involving csc x.

Trigonometric Identity Verification

Verifying trigonometric identities involves substituting equivalent expressions and simplifying both sides to confirm equality. Recognizing standard identities like csc x = 1/sin x allows one to match expressions correctly and complete identity-based problems efficiently.