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Reference

Identity Definition and Examples

Concept

Identity

An identity is an equation that is always true. It may or may not contain variables. If an identity contains variables, no matter what value the variables take, the equation will always hold true.

Rule

Identity Property of Addition

Adding to any number always results in the number itself.

Because of this, is called the Additive Identity.

Proof

Consider a number By the Reflexive Property of Equality, is equal to itself.
Let be another number. If is added to and subtracted from the left-hand side of the above equation, the equality still holds true.
Finally, is equal to
It has been shown that By the Commutative Property of Addition, is also equal to

Rule

Identity Property of Multiplication

Any number multiplied by is equal to the number itself.

Because of this, the number is called the Multiplicative Identity.

Proof

Consider a number By the definition of multiplication, multiplied by another number can be written as times the addition of
If the sum has only one term.
Therefore, is equal to Also, by the Commutative Property of Multiplication,

Rule

Pythagorean Identities

For any angle the following trigonometric identities hold true.

Proof

For Acute Angles
Consider a right triangle with a hypotenuse of
right triangle with hypotenuse 1
By recalling the sine and cosine ratios, the lengths of the opposite and adjacent sides to can be expressed in terms of the angle.
Definition Substitute Simplify

It can be seen that if the hypotenuse of a right triangle is the sine of an acute angle is equal to the length of its opposite side. Similarly, the cosine of the angle is equal to the length of its adjacent side.

Right triangle with hypotenuse 1

By the Pythagorean Theorem, the sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse. Therefore, for the above triangle, the sum of the squares of and is equal to the square of

Since represents a side length, it is not Therefore, by diving both sides of the above equation by the second identity can be obtained.
Simplify

The second identity was obtained.
Since represents a side length, it is not Therefore, by dividing both sides of by the third identity can be proven.
Simplify

Finally, the third identity was obtained.


Proof

For Any Angle

The first identity can be shown using the unit circle and the Pythagorean Theorem. Consider a point on the unit circle in the first quadrant, corresponding to the angle A right triangle can be constructed with

A right triangle with an angle theta on a unit circle
By the Pythagorean Theorem, the sum of the squares of and equals
In fact, this is true not only for points in the first quadrant, but for every point on the unit circle. Recall that, for points on the unit circle corresponding to angle it is known that and that By substituting these expressions into the equation, the first identity can be obtained.
Dividing both sides by either or leads to two variations of the Pythagorean Identity.
Second and third identities are derived from the first one

Rule

Square of a Binomial

When a binomial is squared, the resulting expression is a perfect square trinomial.

For simplicity, depending on the sign of the binomial, these two identities can be expressed as one.

Proof

This rule will be first proven for and then for

This identity can be shown by first rewriting the square as a product.
Multiply parentheses
It has been shown that

In this case, when one term of the binomial is subtracted from the other, the middle term of the perfect square trinomial will instead be negative.
Multiply parentheses
It has been shown that