Applications of PMI in Proving Identities

A tautology is an equation or assertion that is always true, regardless of the circumstances. Sin(x) = 1/cosec(x) is an identity, for example. To “prove” an identity, you must show that one side of the equation can be turned into the other side of the equation using logical procedures. You don’t “prove” anything by plugging values into the identity.

The PMI stands for the Principle of Mathematical Induction (PMI), a mathematical induction method for proving propositions. We can prove the identity valid for all numbers using mathematical induction. An example of such identity is as follows:

1+2+3+⋯+n=n(n+1)/2

More generally, we may prove that a propositional function P(n) is true for all integers n via the application of PMI in proving the identity.

Mathematical induction is a method of proving a given proposition about any well-organized set using mathematics. It’s most commonly used in proving trigonometric identities or establishing claims expressed in n, where n is a natural number. 

Proving the Identity

The technique to prove the identity entails three steps to prove a statement, P(n): They are as follows:

• The first step is to check if the claim is true for simple cases like n = a, i.e. P(a) [Standard Case]
• After this, we have to assume whether the statement is correct for n = k for some k> a, i.e., P(k) [Theory of Induction]
• The last is if the proposition P(n) is true for every n>a and if we assume the truth of P(k) entails the truth of P(k + 1) should also entail

P(k) => P(k + 1) => P(k + 2)…. is true when the base and inductive steps are combined. As a result, P(n) holds for all integers n >a. 

Mathematical induction can be compared to dominoes falling. When a domino falls, it causes the following domino to fall in turn. The first domino topples the second, the second topples the third, and so on. All of the dominoes will be knocked over in the end.

However, some requirements must be met:

• To start the knocking process, the first domino must fall
• This is the primary and essential step
• Any pair of dominoes must have the same space between them Otherwise, a domino could fall without knocking down the next
• The chain of reactions will then come to an end
• Maintaining the same inter-domino distance guarantees that P(k)=> P(k + 1) for each integer k> a
• This is when the inductive process begins

Principle of Mathematical Induction

When any integer x belongs to a class of integers, the successor of x (integer x + 1) likewise belongs. The mathematical induction principle is as follows: 

• If integer 0 belongs to class F and F is hereditary, then any nonnegative integer also belongs to F
• If integer 1 is a member of class F and F is hereditary, then every positive integer is a member of F

The principle is articulated in two ways: once in one form and once in the other. It is not required to distinguish between the two forms of the principle because one can be shown as a result of the other.

The principle is also frequently expressed in the intensional form: A property of integers is hereditary if any integer x has the property, and so does its successor. If a property of the numeral 1 is hereditary, then every positive integer possesses that feature as well.

Proving Trigonometric Identities 

There are numerous methods for proving trigonometric identities; however, some considerations should be made: 

1) Work on the more difficult side and keep it simple.

2) If possible, substitute all trigonometric functions with sinθ and cosθ.

3) Recognize algebraic operations such as factoring and applying the distributive property to reduce the phrase.

4) Use different trigonometric identities. One should keep an eye out for the Pythagorean identity in particular.

5) We should collaborate on all sides.

6) We should keep an eye on the opposite side of the equation and work toward it.

7) Always take into account the “trigonometric conjugate.”

Conclusion

So far, we’ve learned how to prove identities via mathematical induction. In general, mathematical induction can be used to prove a proposition about n. This assertion can be an identity, an inequality, or just a simple vocal remark about n. 

A statement about n can be proved true for all integers n≥a via mathematical induction.

The following steps should be completed:

Verify the statement for n=a in the first step.

Assume that the assertion holds for n=k for some integer in the inductive hypothesis.

At last, use the information gained from the inductive hypothesis to prove that the assertion holds when n=k+1 in the inductive phase.