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.

Identity Without Variables 3=3 [0.8em] Identity With Variables (a-b)(a+b)=a^2-b^2

Rule

Identity Property of Addition

Adding 0 to any number always results in the number itself.

a+0=a

Because of this, 0 is called the Additive Identity.

Proof

Informal Justification

Consider a number a. By the Reflexive Property of Equality, a is equal to itself. a=a Let b be another number. If b is added to and subtracted from the left-hand side of the above equation, the equality still holds true. a=a ⇔ a+b-b=a Finally, b-b is equal to 0. a+b-b=a ⇔ a+0=a ✓ It has been shown that a+0=a. By the Commutative Property of Addition, 0+a is also equal to a.

Rule

Identity Property of Multiplication

Any number multiplied by 1 is equal to the number itself.

a *1=a

Because of this, the number 1 is called the Multiplicative Identity.

Proof

Informal Justification

Consider a number a. By the definition of multiplication, a multiplied by another number n can be written as n times the addition of a. a* n =a+a+... +a_(ntimes) If n=1, the sum has only one term. a* 1 =a_(1time) Therefore, a* 1 is equal to a. Also, by the Commutative Property of Multiplication, 1* a=a.

Rule

Pythagorean Identities

For any angle θ, the following trigonometric identities hold true.

sin^2 θ + cos^2 θ = 1

1 + tan^2 θ = sec^2 θ

1 + cot^2 θ = csc^2 θ

Proof

For Acute Angles

Consider a right triangle with a hypotenuse of 1.

By recalling the sine and cosine ratios, the lengths of the opposite and adjacent sides to ∠ θ can be expressed in terms of the angle.

	Definition	Substitute	Simplify
sin θ	Length of oppositeside to∠ θ/Hypotenuse	opp/1	opp
cos θ	Length of adjacentside to∠ θ/Hypotenuse	adj/1	adj

It can be seen that if the hypotenuse of a right triangle is 1, 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.

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 sin θ and cos θ is equal to the square of 1.

sin^2 θ+ cos^2 θ&= 1^2 ⇓ & sin^2 θ+cos^2 θ&=1

Since cos θ represents a side length, it is not 0. Therefore, by diving both sides of the above equation by cos^2θ, the second identity can be obtained.

sin^2 θ+cos^2 θ=1

DivEqn

.LHS /cos^2θ.=.RHS /cos^2θ.

sin^2 θ+cos^2 θ/cos^2θ=1/cos^2θ

▼

Simplify

WriteSumFrac

Write as a sum of fractions

sin^2 θ/cos^2θ+cos^2 θ/cos^2θ=1/cos^2θ

QuotOne

a/a=1

sin^2 θ/cos^2θ+1=1/cos^2θ

WritePow

Write as a power

sin^2 θ/cos^2θ+1=1^2/cos^2θ

a^m/b^m=(a/b)^m

(sin θ/cosθ)^2+1=(1/cosθ)^2

sin θ/cos θ=tan θ

tan ^2 θ+1=(1/cosθ)^2

1/cos θ=sec θ

tan ^2 θ+1=sec ^2 θ

CommutativePropAdd

Commutative Property of Addition

1+tan ^2 θ=sec ^2 θ

The second identity was obtained.

1+tan ^2 θ=sec ^2 θ

Since sin θ represents a side length, it is not 0. Therefore, by dividing both sides of sin ^2 θ +cos ^2 θ = 1 by sin ^2 θ, the third identity can be proven.

sin^2 θ+cos^2 θ=1

DivEqn

.LHS /sin^2θ.=.RHS /sin^2θ.

sin^2 θ+cos^2 θ/sin^2θ=1/sin^2θ

▼

Simplify

WriteSumFrac

Write as a sum of fractions

sin^2 θ/sin^2θ+cos^2 θ/sin^2θ=1/sin^2θ

QuotOne

a/a=1

1+cos^2 θ/sin^2θ=1/sin^2θ

WritePow

Write as a power

1+cos^2 θ/sin^2θ=1^2/sin^2θ

a^m/b^m=(a/b)^m

1+(cos θ/sinθ)^2=(1/sinθ)^2

cos θ/sin θ=cot θ

1+cot ^2 θ=(1/sinθ)^2

1/sin θ=csc θ

1+cot ^2 θ=csc ^2 θ

Finally, the third identity was obtained.

1+cot ^2 θ=csc ^2 θ

Proof

For Any Angle

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

By the Pythagorean Theorem, the sum of the squares of x and y equals 1. x^2 + y^2 = 1 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 (x,y) on the unit circle corresponding to angle θ, it is known that x = cos θ and that y = sin θ. By substituting these expressions into the equation, the first identity can be obtained.

cos^2 θ + sin^2 θ = 1

Dividing both sides by either cos^2 θ or sin^2 θ leads to two variations of the Pythagorean Identity.