Trig Identities – Addition and Subtraction

Deriving the addition and subtraction trigonometric identities.

We will start with cosine, and use the result to derive the remaining identities.

Proving .

and are represented in the unit circle below.

Remember and

Using the cosine rule and triangle , find

   

   

Using the distance between points, find

   

   

Remember the Pythagorean identity

   

   

Hence

   

(1)  

We can then use this identity to find .

   

   

Remember and

   

(2)  

We can also find

Remember,

   

   

   

(3)  

We can use equation to find

   

   

   

(4)  

And we can use both the sine and cosine identities to find

Remember

   

   

   

   

   

(5)  

and

(6)