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
= π 
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
with 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. 