The Fundamental and Even-Odd Trigonometric Identities

Let's have a recap of the things we've done so far:

1. We found the Pythagorean Identity / Fundamental Theorem of Trigonometry:
2. We defined the tangent:
3. We defined the cotangent:
4. We defined the secant:
5. We defined the cosecant: Let's start with two immediate conclusions. Take a look at the graph of the sine and cosine again, and look for symmetries.
   

You easily see that:

• The sine is odd. It's symmetric with respect to the origin.
• The cosine is even. It has y-axis symmetry.

You have now figured out the even-odd identities:
Remember these. They will make your life easier.

Now, recall that we defined the tangent as:Let's play around with this for a bit, and see where this leads us.
Let's try first squaring the entire expression:Hmm... this seems familiar. Let's use the Pythagorean Identity and substitute the square of the sine on the numerator:This is going somewhere interesting... maybe you've noticed, maybe not. Let's expand that fraction:We immediately get: Do you see it yet?Yup!You have now derived the following identities:
See how easy it is when you take some time to understand much more simplified concepts?
Good job! Take a break, breathe, have a sip of water and rest your eyes. You're doing great.

You will now begin studying The Angle Addition and Subtraction Identities