The Pythagorean trigonometric identity, also called simply the Pythagorean identity, is an identity expressing the Pythagorean theorem in terms of trigonometric functions. Along with the sum-of-angles formulae, it is one of the basic relations between the sine and cosine functions.

The identity is

where ${\displaystyle \sin ^{2}\theta }$ means ${\textstyle (\sin \theta )^{2}}$.

Any similar triangles have the property that if we select the same angle in all of them, the ratio of the two sides defining the angle is the same regardless of which similar triangle is selected, regardless of its actual size: the ratios depend upon the three angles, not the lengths of the sides. Thus for either of the similar right triangles in the figure, the ratio of its horizontal side to its hypotenuse is the same, namely cos θ.

The elementary definitions of the sine and cosine functions in terms of the sides of a right triangle are:

$${\displaystyle {\begin{alignedat}{3}\sin \theta &={\frac {\mathrm {opposite} }{\mathrm {hypotenuse} }}={\frac {b}{c}}\\\cos \theta &={\frac {\mathrm {adjacent} }{\mathrm {hypotenuse} }}={\frac {a}{c}}\end{alignedat}}}$$

The Pythagorean identity follows by squaring both definitions above, and adding; the left-hand side of the identity then becomes

$${\displaystyle {\frac {\mathrm {opposite} ^{2}+\mathrm {adjacent} ^{2}}{\mathrm {hypotenuse} ^{2}}}}$$