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

Verified step by step guidance

Recall the basic trigonometric identities involving cotangent and cosecant. Specifically, remember that \( \cot x = \frac{\cos x}{\sin x} \) and \( \csc x = \frac{1}{\sin x} \).

Express \( \cot x \) in terms of sine and cosine: \( \cot x = \frac{\cos x}{\sin x} \).

Since \( \csc x = \frac{1}{\sin x} \), rewrite \( \sin x \) as \( \frac{1}{\csc x} \). Substitute this into the expression for \( \cot x \): \( \cot x = \cos x \times \csc x \).

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 \( \cos x = \sqrt{1 - \sin^2 x} \).

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 \( \csc x \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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Textbook Question

Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.

(sin θ - cos θ) (csc θ + sec θ)

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Textbook Question

For each expression in Column I, choose the expression from Column II that completes an identity. One or both expressions may need to be rewritten.

cos² x

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

Textbook Question

Verify that each equation is an identity.