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What is the difference between trigonometric identities and Pythagorean identities?
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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.

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

$\tan^2\theta + 1 = \sec^2\theta$

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