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

Verified step by step guidance

1. Recall the basic trigonometric identities involving cotangent and cosecant. Specifically, remember that \( \cot x = \frac{\cos x}{\sin x} \) and \( \csc x = \frac{1}{\sin x} \).
2. Express \( \cot x \) in terms of sine and cosine: \( \cot x = \frac{\cos x}{\sin x} \).
3. Since \( \csc x = \frac{1}{\sin x} \), rewrite \( \sin x \) as \( \frac{1}{\csc x} \). Substitute this into the expression for \( \cot x \): \( \cot x = \cos x \times \csc x \).
4. Use the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \) to express \( \cos x \) in terms of \( \sin x \), and then in terms of \( \csc x \). Since \( \cos^2 x = 1 - \sin^2 x \), take the square root to find \( \cos x = \sqrt{1 - \sin^2 x} \).
5. Replace \( \sin x \) with \( \frac{1}{\csc x} \) in the expression for \( \cos x \), so \( \cos x = \sqrt{1 - \left( \frac{1}{\csc x} \right)^2} \). Finally, substitute this back into \( \cot x = \cos x \times \csc x \) to write \( \cot x \) entirely in terms of \( \csc x \).

Key Concepts

Reciprocal Trigonometric Functions

Reciprocal functions relate pairs of trigonometric functions such as sine and cosecant, cosine and secant, tangent and cotangent. Specifically, cotangent is the reciprocal of tangent, and cosecant is the reciprocal of sine. Understanding these relationships allows rewriting one function in terms of another.

Pythagorean Identity involving Cosecant

The Pythagorean identity states that 1 + cot²x = csc²x. This identity connects cotangent and cosecant, enabling expressions of cotangent in terms of cosecant. It is derived from the fundamental identity sin²x + cos²x = 1 by dividing through by sin²x.

Algebraic Manipulation of Trigonometric Expressions

Transforming trigonometric expressions often requires algebraic skills such as isolating variables, factoring, and taking square roots. To express cot x in terms of csc x, one must rearrange the Pythagorean identity and solve for cot x, considering domain restrictions and the sign of the function.