Core Concepts: The Three Pythagorean Identities

Core Concepts

Learn the primary Pythagorean identity, $\sin^2\theta + \cos^2\theta = 1$, and how to derive the other two involving tangent/secant and cotangent/cosecant. This stack also covers rearranged forms of these identities, which are crucial for substitutions in proofs and simplifications.

Core Concepts: The Three Pythagorean Identities

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What is the fundamental Pythagorean identity involving $\sin(\theta)$ and $\cos(\theta)$?

$$ \sin^2(\theta) + \cos^2(\theta) = 1 $$

How can you rearrange the primary Pythagorean identity to solve for $\sin^2(\theta)$?

Subtract $\cos^2(\theta)$ from both sides of the main identity. $$ \sin^2(\theta) = 1 - \cos^2(\theta) $$

How can you rearrange the primary Pythagorean identity to solve for $\cos^2(\theta)$?

Subtract $\sin^2(\theta)$ from both sides of the main identity. $$ \cos^2(\theta) = 1 - \sin^2(\theta) $$

Show the derivation of the Pythagorean identity involving $\tan(\theta)$ and $\sec(\theta)$.

Start with $\sin^2(\theta) + \cos^2(\theta) = 1$ and divide every term by $\cos^2(\theta)$. This yields: $$ \frac{\sin^2(\theta)}{\cos^2(\theta)} + \frac{\cos^2(\theta)}{\cos^2(\theta)} = \frac{1}{\cos^2(\theta)} $$ Which simplifies to: $$ \tan^2(\theta) + 1 = \sec^2(\theta) $$

State the Pythagorean identity that includes $\tan(\theta)$ and $\sec(\theta)$.

$$ 1 + \tan^2(\theta) = \sec^2(\theta) $$

Rearrange the identity $1 + \tan^2(\theta) = \sec^2(\theta)$ to solve for $\tan^2(\theta)$.

$$ \tan^2(\theta) = \sec^2(\theta) - 1 $$

How is the expression $\sec^2(\theta) - \tan^2(\theta)$ simplified using a Pythagorean identity?

By rearranging $1 + \tan^2(\theta) = \sec^2(\theta)$, we get $\sec^2(\theta) - \tan^2(\theta) = 1$.

Show the derivation of the Pythagorean identity involving $\cot(\theta)$ and $\csc(\theta)$.

Start with $\sin^2(\theta) + \cos^2(\theta) = 1$ and divide every term by $\sin^2(\theta)$. This yields: $$ \frac{\sin^2(\theta)}{\sin^2(\theta)} + \frac{\cos^2(\theta)}{\sin^2(\theta)} = \frac{1}{\sin^2(\theta)} $$ Which simplifies to: $$ 1 + \cot^2(\theta) = \csc^2(\theta) $$

State the Pythagorean identity that includes $\cot(\theta)$ and $\csc(\theta)$.

$$ 1 + \cot^2(\theta) = \csc^2(\theta) $$

Rearrange the identity $1 + \cot^2(\theta) = \csc^2(\theta)$ to solve for $\cot^2(\theta)$.

$$ \cot^2(\theta) = \csc^2(\theta) - 1 $$

How is the expression $\csc^2(\theta) - \cot^2(\theta)$ simplified using a Pythagorean identity?

By rearranging $1 + \cot^2(\theta) = \csc^2(\theta)$, we get $\csc^2(\theta) - \cot^2(\theta) = 1$.

Core Concepts: The Three Pythagorean Identities

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