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. Recognize that \( \tan x = \frac{\sin x}{\cos x} \) and \( \cot x = \frac{\cos x}{\sin x} \).
2. Substitute these identities into the expression: \((\tan x + \cot x)^2 = \left(\frac{\sin x}{\cos x} + \frac{\cos x}{\sin x}\right)^2\).
3. Find a common denominator for the terms inside the parentheses: \(\frac{\sin^2 x + \cos^2 x}{\sin x \cos x}\).
4. Recall the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\), so the expression becomes \(\left(\frac{1}{\sin x \cos x}\right)^2\).
5. Simplify the expression: \(\frac{1}{\sin^2 x \cos^2 x}\), which can be rewritten using the identity \(\csc x = \frac{1}{\sin x}\) and \(\sec x = \frac{1}{\cos x}\) as \(\csc^2 x \sec^2 x\).

Key Concepts

Tangent and Cotangent Functions

The tangent function, denoted as tan(x), is the ratio of the opposite side to the adjacent side in a right triangle, while the cotangent function, cot(x), is the reciprocal of tangent, or the ratio of the adjacent side to the opposite side. Understanding these functions is crucial for manipulating expressions involving them, as they often appear in trigonometric identities and equations.

Algebraic Expansion

Algebraic expansion involves applying the distributive property to multiply expressions, such as (a + b)² = a² + 2ab + b². In the context of the given expression, expanding (tan x + cot x)² will allow us to simplify the result and combine like terms, which is essential for further simplification.

Simplification of Trigonometric Expressions

Simplification of trigonometric expressions involves rewriting them in a more manageable form, often eliminating quotients or combining terms. This process may include using trigonometric identities, such as the Pythagorean identity or reciprocal identities, to express the result in a simpler or more standard form, which is necessary for the final answer.