For each expression in Column I, choose the expression from Column II that completes an identity.
4. cot 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 cotangent in terms of sine and cosine functions: \(\cot x = \frac{\cos x}{\sin x}\).
2. Identify the reciprocal relationship between cotangent and tangent: \(\cot x = \frac{1}{\tan x}\).
3. Recognize that cotangent can also be expressed using cosecant and cosine: \(\cot x = \csc x \cos x\).
4. Review the expressions given in Column II and compare them with these equivalent forms of \(\cot x\).
5. Select the expression from Column II that matches one of these equivalent forms to complete the identity.

Key Concepts

Definition of Cotangent

Cotangent (cot x) is a fundamental trigonometric function defined as the ratio of the adjacent side to the opposite side in a right triangle. It can also be expressed as the reciprocal of the tangent function, i.e., cot x = 1/tan x.

Reciprocal Identities

Reciprocal identities relate trigonometric functions to their reciprocals. For example, cot x is the reciprocal of tan x, and sec x is the reciprocal of cos x. Understanding these identities helps in transforming and simplifying expressions.

Relationship Between Cotangent and Sine/Cosine

Cotangent can also be expressed as the ratio of cosine to sine, cot x = cos x / sin x. This relationship is useful for converting between different trigonometric functions and proving identities.