Step  Image  Equation  Discussion 
1 

and . 
This is the definition of sine and cosine using the angle θ. 
2 

and . 
This is the definition of sine and cosine using the angle θ. 
3 

and . 
This is the definition of sine and cosine using the angle φ. 
4 

and 
Apply the transitive property of equality to equate sin θ with cos φ and cos θ with sin φ. 
5 

α + β + γ = &pi 
This is the Angle Sum Theorem. 
6 


Use the subsitution property of equality to substitute θ for α, φ for β and for γ. 
7 


Apply additive property of equality to add  to both sides of the equation. 
8 


Simplify the equation by combining the constants on both sides of the equation. 
9 


Apply the addition property of equality to add θ to both sides of the equation. 
10 


Cancel θ  θ on the left side of the equation. 
11 

and 
Take the equations from step 4 and apply the subsitution property of equality witht the equations from step 10. These are the first two cofunction identities. 
12 


Use the equations from step 11, and the defintions of tangent and cotangent to get the tangent identity. 
13 


Use the equations from step 11, and the defintions of tangent and cotangent to get the cotangent identity. 
14 


Use the equations from step 11, and the defintions of tangent and cotangent to get the cosecant identity. 
15 


Use the equations from step 11, and the defintions of tangent and cotangent to get the secant identity. 