Trigonometric Identities 

In this chapter, multiple trigonometric identities are discussed. These include reciprocal identities, quotient identities, even/odd identities, co-function identities, and Pythagorean identities. 

Reciprocal Identities:

                   

                   

                       




Quotient Identities:

                    

Even & Odd Functions:
The definition of an even function is that a function is symmetric to the y-axis. For this to be true, the output must always be the same, regardless if the input has the same or opposite sign. Cosine and Secant are the only two functions that fits this definition.

                       
               

           

Co-Function Identities: (*Co-functions of complementary angles are always equal)

                                           

          

           

If you look at the triangle at the right, we can see that that sin(theta) equal to (a/c). Cos [(pi/2)-theta] is also equal to (a/c). Therefore, sin and cosine are referred to as co-function identities. 

Pythagorean Identities:
 

These identities are solely based on the right triangle and the Pythagorean Theorem, hence their name. 

Pythagorean Identity #1

This is how the first identity is derived:
The triangle at the right has a sin(theta) of a/c. Rearranged, (a=csintheta). cos(theta) of b/c. Rearranged, (b=costheta). The Pythagorean Theorem is a² + b² = c²
If we were to substitute sintheta for a and costheta for b, we get the following: 

At this point, we divide by c^2 and we get the first Pythagorean identity listed in the beginning. 

Pythagorean Identity #2

If we divide by cos²ө from the first formula, the cosines will divide out and you will have the inverse of cosine, leaving the second identity:

Pythagorean Identity #3

After dividing the formula above by tangent, we will be left with the inverse and secant over tangent, leaving us with the final Pythagorean identity: