Trig. Identities
The trigonometric fuctions that were introduced in the previous sections are all related to each other through different equations. These are called the TRIGONOMETRIC IDENTITIES.
We will first take a look at the Reciprocal Identities:
csc t = 1/sin t  sec t = 1/cos t  cot t = 1/tan t 
tan t = sin t /cos t  cot t = cos t /sin t 
 Notation:
 sin^{2} t = (sin t)^{2} =
(sin t)(sin t).
sin^{k} t = (sin t)^{k}
This is true for all trigonometric functions.
Next, we will look at the Pythagorean Identities. Recall that we can place a right triangle in the unit circle like so:
Then we get the triangle with the sides and hypotnuse as follows:
Using Pythagorean's Thoerem, we get sin^{2} t + cos^{2} t = 1
Now dividing by sin^{2} t we end up with
1 + cot^{2} t = csc^{2} t.
By doing a similar thing with cos t the result is
tan^{2} t + 1 = sec^{2} t
Here are a few examples using these identity properties.
Another important set of identities are the addition and subtraction formulas. These are only for sine and cosine and are as follows.

sin (s+t) = (sin s)(cos t) +
(cos s)(sin t)
sin (st) = (sin s)(cos t)  (cos s)(sin t)
cos (s+t) = (cos s)(cos t)  (sin s)(sin t)
cos (st) = (cos s)(cos t) + (sin s)(sin t)
The last topic for this section is Double and Half Angle Formulas. These are just as useful as the above addition and subtraction formulas. The doubleangle fromula allow us to find the values of the trig. functions at 2x, and the halfangle formulas relate the values of the trig functions at x/2 to their values at x.
Doubleangle Formulas:
 sin 2x = 2 sin x cos x
cos 2x = cos^{2} x  sin^{2} x
 sin^{2} = (1  cos 2x)/2
cos^{2} = (1 + cos 2x)/2
Let's do a couple of examples to practice this idea before moving to the Exercises.