An identity is an equality that holds for all values where both sides make sense. Demonstrating identities is the great training in manipulating trigonometry: each step uses an already proven relationship. The story To demonstrate identities it is necessary to write the functions in short form. Until the 17th century this did not exist: each relationship was said in words — “the sine of the complement of the arc”, “the tangent of the bent arc” — and a one-line identity occupied a paragraph. The abbreviations appeared little by little: Gunter used sin in a figure in 1624, the Frenchman Pierre Hérigone printed it in a book in 1634, and the Englishman William Oughtred wrote something like s co arc for cosine. It was Euler, in Introductio (1748), who established the use of sin., cos., tang. as quantities that are added, multiplied and squared — and it is this use that allows you to manipulate an identity as an algebraic expression.
In previous posts we defined porpositions, theorems, corollaries, among others. To demonstrate theorems, it means that the given proposition is a tautology, in which we obtain through the truth tables. Below, we present some methods for demonstrating results. Deductive method The deductive method is obtained through the use of definitions and other pre-established results. For this, we use the implications $(\Rightarrow)$ and logical bi-implications $(\Leftrightarrow)$. Example 1: With the following statements, we will show whether it is a theorem or not. For this, the following hypotheses are: $H_1$: time is money. $H_2$: bum has a lot of time. And the thesis: $T$: Tramp has a lot of money. Is the proposition $H_1 \wedge H_2 \Rightarrow T$ a theorem?