# Pythagorean trigonometric identity

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The Pythagorean trigonometric identity is a trigonometric identity based on the application of the Pythagorean Theorem to the unit triangle.

The fundamental identity is:

Identity: $\sin^2(\theta)+\cos^2(\theta)=1$ , for all $\theta\in\R$
Theorem. '
Prerequisites:
Pythagorean theorem: $a^2+b^2=c^2$

Proof. Assume a point $C$ in the unit circle centered at the origin forming an angle $\theta$ . Let $B$ be a point on the x-axis corresponding to the x-coordinate of $C$ . Let $A$ be the origin. A right triangle would be formed: $\triangle ABC$ . The length of $\overline{AB}$ would be equal to $\cos(\theta)$ and the length of $\overline{BC}$ would be equal to $\sin(\theta)$ . The length of $\overline{AC}$ is 1 (since we're using the unit circle). Using the Pythagorean theorem, we have: $\sin^2(\theta)+\cos^2(\theta)=1$

## Implications

Due to this fundamental relationship, other Pythagorean Identities emerge through the use of:

• the complimentary and cofunction properties
• the reciprocal functions
• the quotient identities

The other identities include:

Identity: $\tan^2(\theta)+1=\sec^2(\theta)$
Theorem. '
Proof.
$\sin^2(\theta)+\cos^2(\theta)=1$
$\frac{\sin^2(\theta)}{\cos^2(\theta)}+\frac{\cos^2(\theta)}{\cos^2(\theta)}=\frac{1}{\cos^2(\theta)}$
$\tan^2(\theta)+1=\sec^2(\theta)$

Identity: $1+\cot^2(\theta)=\csc^2(\theta)$
Theorem. '
Proof.
$\sin^2(\theta)+\cos^2(\theta)=1$
$\frac{\sin^2(\theta)}{\sin^2(\theta)}+\frac{\cos^2(\theta)}{\sin^2(\theta)}=\frac{1}{\sin^2(\theta)}$
$1+\cot^2(\theta)=\csc^2(\theta)$