differentiable, so that in fact it subsumes the previous two. Proof using the differential equation Sine and cosine can be defined as the two solutions to the differential equation: satisfying respectively and . It follows from the theory of ordinary differential equations that the first solution, sine, has the second, cosine, as its derivative, and it follows from this that the derivative of cosine is the negative of the sine. The identity is equivalent to the assertion that the function is constant and equal to 1. Differentiating using the chain rule gives: so z is constant. A calculation confirms that z(0) = 1, and z is a constant so z = 1 for all x, so the Pythagorean identity is established. A similar proof can be completed using power series as above to establish that the sine has as its derivative the cosine, and the cosine has as its derivative the negative sine. In fact, the definitions by ordinary differential equation and by power series lead to similar derivations of most identities. This proof of the identity has no direct connection with Euclid's demonstration of the Pythagorean theorem. Proof using Euler's formula Euler's formula states that So, See also Pythagorean theorem List of trigonometric identities Unit circle Power series Differential equation Notes Mathematical identities Articles containing proofs Trigonometry Identity