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.
cot² θ(1 + tan² θ)

Guida verificata passo dopo passo

Recall the definitions of cotangent and tangent in terms of sine and cosine: \(\cot \theta = \frac{\cos \theta}{\sin \theta}\) and \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).

Rewrite the given expression \(\cot^{2} \theta (1 + \tan^{2} \theta)\) by substituting these definitions: \(\left(\frac{\cos \theta}{\sin \theta}\right)^{2} \left(1 + \left(\frac{\sin \theta}{\cos \theta}\right)^{2}\right)\).

Simplify inside the parentheses first: \(1 + \left(\frac{\sin \theta}{\cos \theta}\right)^{2} = 1 + \frac{\sin^{2} \theta}{\cos^{2} \theta}\).

Combine the terms inside the parentheses over a common denominator: \(\frac{\cos^{2} \theta}{\cos^{2} \theta} + \frac{\sin^{2} \theta}{\cos^{2} \theta} = \frac{\cos^{2} \theta + \sin^{2} \theta}{\cos^{2} \theta}\).

Use the Pythagorean identity \(\sin^{2} \theta + \cos^{2} \theta = 1\) to simplify the numerator, then multiply by the squared cotangent term and simplify the entire expression so that no quotients remain and all functions are in terms of \(\theta\) only.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Trigonometric Ratios and Their Definitions

Understanding the basic trigonometric functions—sine, cosine, tangent, and cotangent—is essential. Tangent is defined as sine divided by cosine (tan θ = sin θ / cos θ), and cotangent is its reciprocal (cot θ = cos θ / sin θ). Expressing all functions in terms of sine and cosine allows for easier manipulation and simplification.

Video consigliato:

Percorso guidato

6:04

Introduction to Trigonometric Functions

Pythagorean Identity

The Pythagorean identity states that sin² θ + cos² θ = 1. This fundamental relationship helps simplify expressions involving squares of trigonometric functions. For example, the identity 1 + tan² θ = sec² θ can be rewritten using sine and cosine to aid in simplification.

Video consigliato:

Percorso guidato

6:25

Pythagorean Identities

Algebraic Simplification of Trigonometric Expressions

After rewriting expressions in terms of sine and cosine, algebraic techniques such as factoring, combining like terms, and eliminating quotients are used to simplify the expression. The goal is to write the expression without fractions and only in terms of sine and cosine functions of θ.

Video consigliato:

Percorso guidato