# Proof: Pythagorean Identity

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## Proof

### Prerequisites

- The antiderivative of 0 is a constant
- Series definition of sine and cosine (in particular and )
- Differential of is , differential of is
- Linearity of the derivative, the Chain rule

### Proof

(linearity of the derivative)

(chain rule)

(evaluating the differentials)

As the derivative of the expression is zero, this implies for some constant k. Evaluating at , which means , implying

## Geometric "proof"

It is possible to use geometry to prove the statement, however it only holds for

### Prerequisites

- Pythagorean Theorem: , in any right triangle. - [ Proof ]
- The definition of the trigonometric functions as ratios of the sides of a right triangle:

### Proof

Given an arbitrary right triangle, the following are true:

Here,

And therefore,

Via the Pythagorean Theorem, the legs (here: opposite and adjacent) are "a," and "b" where hypotenuse is "c".

So,