Step-by-step examples for proving Pythagorean identities

📚 Quick Study Guide

• 📐 The Pythagorean identity is: $\sin^2(\theta) + \cos^2(\theta) = 1$.
• 🔄 We can manipulate this identity to get: $\sin^2(\theta) = 1 - \cos^2(\theta)$ and $\cos^2(\theta) = 1 - \sin^2(\theta)$.
• ➗ Dividing the original identity by $\cos^2(\theta)$ gives: $\tan^2(\theta) + 1 = \sec^2(\theta)$.
• ➕ Similarly, dividing by $\sin^2(\theta)$ gives: $1 + \cot^2(\theta) = \csc^2(\theta)$.
• 💡 When proving identities, start with the more complex side and simplify it to match the other side.
• ✍️ Use algebraic manipulations and other trigonometric identities to simplify.
• 🧐 Remember definitions: $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$, $\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}$, $\sec(\theta) = \frac{1}{\cos(\theta)}$, $\csc(\theta) = \frac{1}{\sin(\theta)}$.

Practice Quiz

1. What is the fundamental Pythagorean identity?
   1. $\sin(\theta) + \cos(\theta) = 1$
   2. $\sin^2(\theta) + \cos^2(\theta) = 1$
   3. $\tan^2(\theta) + \cot^2(\theta) = 1$
   4. $\sec^2(\theta) + \csc^2(\theta) = 1$
2. Which of the following is equivalent to $\sin^2(\theta)$?
   1. $1 + \cos^2(\theta)$
   2. $1 - \cos^2(\theta)$
   3. $\cos^2(\theta) - 1$
   4. $\frac{1}{\cos^2(\theta)}$
3. What identity do you get when you divide $\sin^2(\theta) + \cos^2(\theta) = 1$ by $\cos^2(\theta)$?
   1. $1 + \cot^2(\theta) = \csc^2(\theta)$
   2. $\tan^2(\theta) - 1 = \sec^2(\theta)$
   3. $\tan^2(\theta) + 1 = \sec^2(\theta)$
   4. $1 + \tan^2(\theta) = \csc^2(\theta)$
4. $\tan^2(\theta)$ is equal to which of the following?
   1. $\sec^2(\theta) + 1$
   2. $\sec^2(\theta) - 1$
   3. $1 - \sec^2(\theta)$
   4. $\csc^2(\theta) - 1$
5. Which expression is equivalent to $\csc^2(\theta) - 1$?
   1. $\tan^2(\theta)$
   2. $\sec^2(\theta)$
   3. $\cot^2(\theta)$
   4. $\sin^2(\theta)$
6. Simplify the expression: $\frac{\sin^2(\theta)}{1 - \cos^2(\theta)}$
   1. $\cos^2(\theta)$
   2. $\sin^2(\theta)$
   3. $1$
   4. $\tan^2(\theta)$
7. Which of the following is NOT a Pythagorean identity?
   1. $\sin^2(\theta) + \cos^2(\theta) = 1$
   2. $1 + \cot^2(\theta) = \csc^2(\theta)$
   3. $\tan^2(\theta) + 1 = \sec^2(\theta)$
   4. $\sin^2(\theta) - \cos^2(\theta) = 1$