In mathematics, an identity is a relationship that is true for all possible values.

To determine the difference between trigonometric identities and pythagorean identities, it is important to define them first.

A **trigonometric** **identity** is an equation that involves trigonometric functions and is true for every single value substituted for the variable.

The Pythagorean Theorem is the relationship between the hypotenuse, opposite and adjacent sides of a right-angled triangle.

\[x^2+y^2=r^2\] ..... (1)

where $r$ is the hypotenuse.

Now using the trigonometric relationships of the associaed angles in the given right-angled triangles, we can deduce the following ratios:

\[\sin{\theta}=\dfrac{y}{1}\]

and

\[\cos{\theta}=\dfrac{x}{1}\]

and

\[\tan{\theta}=\dfrac{y}{x}\]

Using (1), we have trigonometric identities that involve the Pythagorean Theorem. Examples are:

\[\sin^2 \theta + \cos^2 \theta = 1\]

and

\[\dfrac{\sin^2\theta}{\cos^{2}\theta}+\dfrac{\cos^2\theta}{\cos^2\theta} = \dfrac{1}{\cos^2\theta}\]

and

\[\tan^2\theta + 1 = \sec^2\theta\]