Pythagorean trigonometric identity

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 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 $\overline{B C}$ 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
$\tan^2 \theta + 1 = \sec^2 \theta$

Identity
$1 + \cot^2 \theta = \csc^2 \theta$

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

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