Sections:

Trigonometric Identities and Equations, Page 5

Pythagorean Identities

The next set of identities is called Pythagorean Identities.

For all values of theta, (where defined), the following are true:

sin^2 (theta) + cos^2 (theta) = 1; tan^2 (theta) + 1 = sec^2 (theta); 1 + cot^2 (theta) = csc^2 (theta)

Again, in order to look more closely at sin^2 (theta) + cos^2 (theta) = 1, you need to look at the Unit Circle.

5.3.5 graphic 2

Using the Pythagorean Theorem, y2 + x2 = 1. Remember that the x-value is also equal to costheta and the y-value is equal to sintheta.

You can rewrite the equation as sin^2 (theta) + cos^2 (theta) = 1.

You can also use sin^2 (theta) + cos^2 (theta) = 1 to find tan^2 (theta) + 1 = sec^2 (theta).

Take sin^2 (theta) + cos^2 (theta) = 1 and divide by cos2theta.

sin^2 (theta)/cos^2 (theta) + cos^2 (theta)/cos^2 (theta) = 1/cos^2 (theta); tan^2 (theta) + 1 = sec^2 (theta)