Perform each transformation. See Example 2.
Write cot x in terms of csc x.

Verified step by step guidance

1. Start with the identity for cotangent: \( \cot x = \frac{1}{\tan x} \).
2. Recall the identity for tangent in terms of sine and cosine: \( \tan x = \frac{\sin x}{\cos x} \).
3. Substitute the tangent identity into the cotangent expression: \( \cot x = \frac{1}{\frac{\sin x}{\cos x}} = \frac{\cos x}{\sin x} \).
4. Use the identity for cosecant: \( \csc x = \frac{1}{\sin x} \).
5. Substitute the cosecant identity into the cotangent expression: \( \cot x = \cos x \cdot \csc x \).

Key Concepts

Reciprocal Trigonometric Functions

In trigonometry, reciprocal functions are pairs of functions that are inversely related. For example, the cosecant (csc) is the reciprocal of the sine (sin), and the cotangent (cot) is the reciprocal of the tangent (tan). Understanding these relationships is crucial for transforming one trigonometric function into another, such as expressing cot x in terms of csc x.

Cosecant Function

The cosecant function, denoted as csc x, is defined as the reciprocal of the sine function. Mathematically, csc x = 1/sin x. This function is particularly important when working with angles in a right triangle, as it relates to the ratio of the hypotenuse to the opposite side. Recognizing how csc x interacts with other trigonometric functions is essential for transformations.

Cotangent Function

The cotangent function, represented as cot x, is defined as the ratio of the adjacent side to the opposite side in a right triangle, or cot x = cos x/sin x. It can also be expressed in terms of sine and cosecant, specifically as cot x = 1/tan x. Understanding how cotangent relates to sine and cosecant is key to performing the required transformation in the question.