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 $\theta \in \mathbb {R}$

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. $\sin ^{2}(\theta )+\cos ^{2}(\theta )=1$ ${\frac {\sin ^{2}(\theta )}{\sin ^{2}(\theta )}}+{\frac {\cos ^{2}(\theta )}{\sin ^{2}(\theta )}}={\frac {1}{\sin ^{2}(\theta )}}$ $1+\cot ^{2}(\theta )=\csc ^{2}(\theta )$