Using the Pythagorean Identities .............................

More Simplifying

sin²θ + cos²θ = 1
sin²θ = 1 - cos²θ
cos²θ = 1 - sin²θ

tan²θ + 1 = sec²θ
tan²θ = sec²θ - 1
sec²θ - tan²θ = 1

1 + cot²θ = csc²θ
cot²θ = csc²θ - 1
csc²θ - cot²θ = 1

Simplifying Expressions

Example 3:

Start by changing everything to sin's and cos's:

Multiply the last term:

Now make the first term into a fraction, so we can add the terms:

and add them ...

Finally, use a Pythagorean identity, and simplify:

Example 4:

There are a number of things you could do here.
Experience will tell you that when you can factor, you should do that:

Now we can change everything to sin's and cos's:

Combining the terms in the bracket will require they both be fractions:

Finally, reduce and simplify:

Next we'll prove some other identities. Move on to page four ...

Intro | Simplifying | More Simplifying | Verifying

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