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

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,

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