Simplify each of the following so that it is expressed exclusively in terms of sine, cosine or the number 1.

Simplify each of the following so that it is expressed exclusively in terms of sine, cosine or the number 1.

Simplify each of the following so that it is expressed exclusively in terms of sine, cosine or the number 1.

Simplify each of the following so that it is expressed exclusively in terms of sine, cosine or the number 1.

Simplify each of the following so that it is expressed exclusively in terms of sine, cosine or the number 1.

The Pythagorean Identities ...

The corollaries ...

Which implies ...

Simplify each of the following so that it is expressed exclusively in terms of sine, cosine or the number 1.

Simplify each of the following so that it is expressed exclusively in terms of sine, cosine or the number 1.

Let's warm up ...

Simplify each of the following as much as possible:

Prove each of the following identities ...

Prove each of the following identities ...

Prove each of the following identities ...

Some Strategies for Proving Trigonometric Identities

(1) Work with the more complicated side of the identity first.

(2) Rewrite both sides of the identity exclusively in terms of sine and cosine.

(3) Use a Pythagorean identity to make an appropriate substitution.

(4) Simplify complex fractions or rewrite fractions sums or differences with a single denominator.

(5) Use factoring (especially differences of squares).

All of the above are just suggestions or "rules of thumb." Feel free to disregard any or all of the above at any time.

Even and Odd Identities

sin(-x) = -sin(x) cos(-x) = cos(x) tan(-x) = -tan(x)

The sine and tangent functions are ODD functions. Cosine is an EVEN function.

Prove each of the following identities ...

Prove each of the following identities ...

Prove each of the following identities ...