2. The synthesis method is based on the known facts (known conditions, important inequalities or proven inequalities), with the help of the properties of inequalities and related theorems, and through gradual logical reasoning, the inequalities to be proved are finally derived. Its characteristics and ideas are "from cause to effect", from "known" to "need to know" and gradually deduce "conclusion" 3. Analytic method is to analyze the sufficient condition of this inequality from the inequality that needs to be proved, and then transform it into judging whether that condition is met. Its characteristic and thought is "depending on the fruit", that is, from "unknown" to "known". 4. The proof of some inequalities by reduction to absurdity is not clear from the positive proof, but it can be considered from the positive and negative perspectives, that is, to prove inequality A>B, first assume that A≤B, and deduce contradictions from topics and other properties, thus affirming that the inequalities involved in A> B are negative propositions, unique propositions or contain "at most", "at least", "nonexistent" and "impossible".
5. method of substitution and method of substitution introduced one or more variables to replace some inequalities with complex structure, many variables and unclear relations among variables, thus simplifying the original structure or realizing some transformation and adaptation, and bringing new enlightenment and methods to the proof. There are two main forms of substitution. (1) Triangular substitution method: it is often used to prove conditional inequalities. When the given conditions are complicated and one variable cannot be easily expressed by another variable, we can consider triangle substitution and use the same parameter to express two variables. If this method is used properly, it can communicate the relationship between trigonometry and algebra and transform complex algebraic problems into trigonometric problems. According to specific problems, the triangle substitution method is as follows: ① If x2+y2= 1, let x=cosθ and y = sin θ; ② If x2+y2≤ 1, x=rcosθ, y = rsin θ (0 ≤ r ≤1); (3) For the inequality involved, because |x|≤ 1, we can set x = cos θ; (4) If x+y+z=xyz, we can know from tanA+tanB+tanC=tanAtan-BtanC that x=taaA, y=tanB, z=tanC, where A+B+C=π. (2) Incremental substitution method: in the symmetrical formula (two letters can be exchanged arbitrarily, and the algebraic formula remains unchanged) and the given alphabetical order (such as A >；; B>c etc. ), consider changing elements by incremental method, with the aim of reducing elements by changing elements, making the problem difficult and easy, and simplifying the complex. For example, a+b= 1 can be substituted by a= 1-t, b=t or a= 1/2+t, b =1/2-t.
6. Scaling method Scaling method is to prove inequality A.