Pythagorean Identity

How to use the Pythagorean identity to solve the related problems: formula, 3 examples, and their solutions.

Formula

Formula

sin2 θ + cos2 θ = 1
is the Pythagorean identity.

As you can see,
this identity looks like the Pythagorean theorem:
a2 + b2 = c2.

Derived Formulas

These are the formulas (identities)
derived from sin2 θ + cos2 θ = 1.

sin2 θ + cos2 θ = 1
Move +cos2 θ to the right side.
Then sin2 θ = 1 - cos2 θ.

sin2 θ + cos2 θ = 1
Move sin2 θ to the right side.
Then cos2 θ = 1 - sin2 θ.

These two formulas are used
when changing sine to cosine
or changing cosine to sine.

sin2 θ + cos2 θ = 1
Divide both sides by cos2 θ.
Then tan2 θ + 1 = sec2 θ.

This formula is used
when changing tangent to secant.

sin2 θ + cos2 θ = 1
Divide both sides by sin2 θ.
Then 1 + cot2 θ + 1 = csc2 θ.

This formula is used
when changing cotangent to cosecant.

Example 1

Example

Solution

To show that the given equation is an identity,

start from the left side,
then derive the right side.

First write the left side
(sin θ + cos θ)2/(sin θ).

(sin θ + cos θ)2
= sin2 θ + 2 sin θ cos θ + cos2 θ

Square of a Sum: (a + b)2

sin2 θ + cos2 θ = 1

Split the function into two functions.

1/(sin θ) = csc θ

Cosecant: in a Right Triangle

(2 sin θ cos θ)/(sin θ) = 2 cos θ

Let's see what you've solved.
You changed the left side, (sin θ + cos θ)2/(sin θ),
to the right side, csc θ + 2 cos θ.

So the given equation is an identity.

So write
∴ (sin θ + cos θ)2/(sin θ) = csc θ + 2 cos θ.

This is the solution of this example.

Example 2

Example

Solution

cos2 θ = 1 - sin2 θ

1 - sin2 θ
= 12 - sin2 θ
= (1 + sin θ)(1 - sin θ)

Factor the Difference of Two Squares: a2 - b2

Cancel the common factors (1 - sin θ).

Then 1 + sin θ is the answer.

Example 3

Example

Solution

First write the left side
sin2 θ(1 + tan2 θ).

1 + tan2 θ = sec2 θ

Secant is the reciprocal of cosine.

So sec2 θ = 1/(cos2 θ).

Combine sin2 θ and 1/(cos2 θ).

(sin θ)/(cos θ) = tan θ

Quotient Identity

So (sin2 θ)/(cos2 θ) = tan2 θ.

Let's see what you've solved.
You changed the left side, sin2 θ(1 + tan2 θ),
to the right side, tan2 θ.

So the given equation is an identity.

So write
∴ sin2 θ(1 + tan2 θ) = tan2 θ.

This is the solution of this example.