The Pythagorean Theorem works on right triangles. If you consider the

An observant student may guess that other Pythagorean identities exist with the rest of the trigonometric functions. Is

### Pythagorean Identities

The proof of the **Pythagorean identity** for sine and cosine is essentially just drawing a right triangle in a unit circle, identifying the cosine as the

Most people rewrite the order of the sine and cosine so that the sine comes first.

The two other Pythagorean identities are:

1+cot2x=csc2x tan2x+1=sec2x

To derive these two Pythagorean identities, divide the original Pythagorean identity by

To derive the Pythagorean identity

Similarly, to derive the Pythagorean identity

### Examples

#### Example 1

Earlier, you were asked if

Visually, the right triangle connecting tangent and secant can also be observed in the unit circle. Most people do not know that tangent is named “tangent” because it refers to the distance of the line tangent from the point on the unit circle to the

#### Example 2

Simplify the following expression:

Note that factoring the Pythagorean identity is one of the most powerful applications. This is very common and is a technique that you should feel comfortable using.

#### Example 3

Prove the following trigonometric identity:

Group the terms and apply a different form of the second two Pythagorean identities which are

#### Example 4

Simplify the following expression.

#### Example 5

Simplify the following expression.

Note that initially, the expression is not the same as the Pythagorean identity.

### Review

Prove each of the following:

1.

2.

3.

4.

5.

6.

Simplify each expression as much as possible.

7.

8.

9.

10.

11.

12.

13.

14.

15. \begin{align*}\frac{1-\sin^2 x}{\cos x}\end{align*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 6.2.