Perform each indicated operation and simplify the result so that there are no quotients.
(tan x + cot x)²

Verified step by step guidance

1. Start by writing the expression explicitly: \( (\tan x + \cot x)^2 \).
2. Recall that \( \tan x = \frac{\sin x}{\cos x} \) and \( \cot x = \frac{\cos x}{\sin x} \). Substitute these into the expression: \( \left( \frac{\sin x}{\cos x} + \frac{\cos x}{\sin x} \right)^2 \).
3. Find a common denominator for the terms inside the parentheses, which is \( \sin x \cos x \), and rewrite the sum as a single fraction: \( \left( \frac{\sin^2 x + \cos^2 x}{\sin x \cos x} \right)^2 \).
4. Use the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \) to simplify the numerator, so the expression becomes \( \left( \frac{1}{\sin x \cos x} \right)^2 \).
5. Square the fraction to get \( \frac{1}{\sin^2 x \cos^2 x} \). To eliminate quotients, rewrite this as \( \csc^2 x \sec^2 x \), since \( \csc x = \frac{1}{\sin x} \) and \( \sec x = \frac{1}{\cos x} \).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. They allow simplification and transformation of expressions, such as converting quotients into products or sums, which is essential for simplifying (tan x + cot x)².

Pythagorean Identity

The Pythagorean identity states that sin²x + cos²x = 1. This fundamental relation helps express tangent and cotangent in terms of sine and cosine, enabling the simplification of expressions involving tan x and cot x by rewriting them without quotients.

Algebraic Expansion and Simplification

Expanding expressions like (tan x + cot x)² involves applying the distributive property (a + b)² = a² + 2ab + b². After expansion, combining like terms and using trigonometric identities helps eliminate quotients and simplify the expression to a more manageable form.