Deriving Pythagorean Identities

First, we set up a right triangle inside the unit circle. We label the sides appropriately. In a triangle, we know that sine = y/r and cosine = x/r
X is the adjacent side of the triangle, while Y is usually the opposite side, and the R equals 1.
With this information, we can substitute in values, where sin = y/1 = y and cos = x/1 = x

Now thinking back to the Pythagorean Theorem, we know that it says if you square the sides of the triangle, add them, we end up with the hypotenuse. In other words (or variables, I should say), x^2+y^2=r^2

NOW REPLACE THE VARIABLES!

cos^2+sin^2 = 1


It has now become one of the three Pythagorean identities, the other two can be easily derived from this one. Let's have a look now.

The next identity is 1+tan^2theta=sec^2theta

So how do we get this? Well, take the first formula and divide it by cosine. The first part should look something like sin^2/cos^2 (THIS IS Y/X, AKA, TANGENT). The second part is cos over cos, which is equal to 1. The last part is 1/cos (THIS IS SECANT!)
So now we simplify it and get:
1+tan^2theta=sec^2theta

Pretty simple right?
Now I will show you how to derive the third identity!

Goal to make
1+tan^2theta=sec^2theta
look like
1+cot^2theta=csc^2theta





This is accomplished by going back to the first identity and instead of dividing by cosine, we will divide by sine.

Following the same steps as before, we should end up with sine over sine (1) and cosine over sine (RECOGNIZE THIS AS COTANGENT) and 1 over sine (RECOGNIZE THIS AS COSECANT)
Now we simplify, and end up with:

1+tan^2theta=sec^2theta