By Menger K.
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J) IT ES? Substituting i *j Into the equality, we see that tan -sin/cos. 2 Other useful identities are sin2* cos 2 - 1 and 2 1 + tan *2rec cos . We postulate an Inverse of tan on both sides and call it arctan. 30 II. 1. THE ALGEBRA OP CALCULUS The Algebra of Derivatives. We shall now introduce an operator D associating with a function f a function Df, called the derivative of f• We shall not attempt to formulate criteria as to which functions form the domain of the operator D or as to which constants, if any, may be substituted into Df.
Substitution will be assumed to satisfy the following lawst I. Associative Law. ,ha )]. For some purposes it is convenient to denote the a± functions substituted into gA by \v***±a. ,r). ,j) - f. III. Ljw^of jpep£esslon. • In the classical notation, a function admitting the suppression of the Index 1 is one which does not depend upon its 1-th variable, as f(x,y,z) = 4*x + 5*log z does not depend upon y. *gr) no matter which functions g^> •••jgp we substitute, then we can suppress any r-1 of the indices and thus arrive at a constant function.
5. The Derivation of the Trigonometric Functions, Let tan be a tangential function, c a constant. -»-tan c«tan c)«rec(l - tan*tan c)2. Substituting 0 we obtain D tan c * D tan 0*(l+tan c*tan c)»rec(l-tan 0*tan c) . 38 Since tan 0 = 0 we have D tan c =D tan 0«(l+tan c*tan c) for each eonatant c. If we have a base of constant a, then D tan - D tan O-U+tan2). We shall now postulate that there Is a tangential function tan for which D tan 0 = 1. From now on we shall reserve the sym^ bol tan for this function given by the postulates 1.