Trigonometry is abundant with identities—they provide important renaming tools
when working with trigonometric expressions.
Three of the most basic trigonometric identities are
discussed in this section, after a quick review of the word ‘identity’.
the Pythagorean Identity:  $\,\sin^2 t + \cos^2 t = 1\,$ 
cosine is an even function:  $\,\cos (t) = \cos t\,$ 
sine is an odd function:  $\,\sin (t) = \sin t\,$ 
An identity is a mathematical sentence that is always true.
Strictly speaking, ‘1 + 1 = 2’ is an identity.
However, the word ‘identity’ is typically reserved for a sentence with one or more variables, that is true
for every possible choice of variable(s).
For example, $\,(x+y)^2 = x^2 + 2xy + y^2\,$ is an identity from algebra.
No matter what real numbers are chosen for $\,x\,$ and $\,y\,$, the equation is true.
For example, letting $\,x = 2\,$ and $\,y = 3\,$ gives:
In this case, it is a simple application of FOIL to see why the equation is always true: $$\cssId{s22}{(x+y)^2 = (x+y)(x+y) = x^2 + 2xy + y^2}$$ However, identities are not always this obvious—it is not always this easy to prove that a given sentence is an identity!
Identities can provide tremendous renaming power.
For example, the identity above allows the expressions $\,(x+y)^2\,$ and $\,x^2 + 2xy + y^2\,$
to be substituted,
one for the other, whenever it is convenient to do so.
In particular, renaming $\,x^2 + 2xy + y^2\,$ as $\,(x+y)^2\,$ (a perfect square) shows that
it is always nonnegative.
The sum $\,x^2 + 2xy + y^2\,$ doesn't readily reveal that it can't ever be negative—but the perfect square $\,(x+y)^2\,$ does.
Different names can reveal different properties of numbers!
The Pythagorean Identity, $\,\sin^2 t + \cos^2 t = 1\,$, is perhaps the most used and most famous trigonometric identity.
Remember—when a mathematical result is given a special name, there's a reason!
The Pythagorean Identity follows immediately from the unit circle definition of sine and cosine:


Look at the green triangle shown at right, in the first quadrant, in the unit circle:
The Pythagorean Identity! 

Recall that even functions have the property that when inputs are opposites, outputs are the same:


By definition:

Recall that odd functions have the property that when inputs are opposites, outputs are also opposites:


By definition:

On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. 
PROBLEM TYPES:
