Pythagorean Identity and why is it necessary/

Question: What is the Pythagorean Identity and why is it necessary

Pythagorean Identities definition:

It is known that any point on a unit circle can be represented by sin θ and cos θ making it possible to draw a triangle with one side the length of  sin θ and another the length of  cos θ and the last side which will naturally be 1 by definition.  With this description the first or original Pythagorean identity is sin^2 θ+cos^2 θ=1.  As you can see this equation kind of resembles the Pythagorean theorem which is a^2+b^2= c^2, because the Pythagorean Identity is the application of the Pythagorean theorem but to a unit circle as described in the picture to the left.

The three Pythagorean Identities: 

The first Pythagorean Identity is as we mentioned before  sin^2θ+cos^2θ=1.  The second Pythagorean theorem  of Tan^2θ +1= sec^2θ and the third Pythagorean identity is 1+cot^2θ=csc^2θ.

Importance of Pythagorean identities and why this name is appropriate:      

The reason why the Pythagorean identities is important is because if you are working on a problem using the unit circle and had the value of a trig function such as cosine but wanted to find sine you will use either of the three Pythagorean Identities.   This identity is important because it sets an expression involving trig functions equal to 1, and this simplification is very helpful for solving equations. As such, this is probably one of the most frequently used trig identities. The word Pythagorean is used to describe the relationship between a^2+b^2=c^2 which applies to the right angle, while the word identity when applying to math means a relation that is true in every interpretation.  So together the word Pythagorean Identity means the relationship between a^2+b^2=c^2 that is true in every interpretation.  This definition and importance of the Pythagorean Identity shows the importance of this name in the way that the Pythagorean Identity has a identity of sin^2θ+cos^2θ=1 which is similar to a^2+b^2=c^2, because in a^2+b^2=c^2 it applies to a right triangle by itself while the Pythagorean Identities applies to unit circles which are different interpretations of each other.  Another example of how this definition applies to how appropriate the name Pythagorean Identity is all of the Pythagorean Identities are different forms of each other.