Deriving Sum - Difference, Two Angles Formula

Category: Trigonometry, Plane Geometry

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Let's consider the given figure:
 

Photo by Math Principles in Everyday Life

Note: Before you proceed with the derivation, you must know the principles of Plane Geometry very well first because you will consider some proving and reasoning at the given figure. 

Solution: 

First, from the x-axis and y-axis, let's draw line 1 and line 2 that will pass at the origin and let's assign point A there. Let ϕ is the measurement of an angle from line 1 to line 2 while θ is the measurement of an angle from line 2 to x-axis. Next, assign point B at line 1. From point B, draw a vertical line that is perpendicular to x-axis and let's assign point E at the x-axis. Consider ΔABE,

In this derivation, we have to eliminate the line segments and we need the angles only. So, we have to do further derivations. From point B, draw a line that is perpendicular to line 2 and let's assign point C at line 2. From point C, draw a vertical line that is perpendicular to the x-axis and let's assign point D at the x-axis. From point C also, draw a horizontal line which is perpendicular to y-axis and let's assign point F at line EB. Consider ΔACD,

                           or

                           or

Consider ΔABC,

                           or

                           or

                  
Consider ΔBCF,

                           or 

           

                           or

Therefore,
              

                

but
             

                      
(Reason: If line FC is parallel to line ED and line FE is parallel to line CD, then the line segment FC is congruent to line segment ED and the line segment EF is congruent to line segment CD. The resulting figure EFCD is a parallelogram. Since each four lines are perpendicular to each other, then the parallelogram is a rectangle.)

          

             

 

If you will substitute -ϕ to the above equation, then

            
Let's have another one,

  
using the same figure, we have

     
but
          

      
(Reason: If line FC is parallel to line ED and line FE is parallel to line CD, then the line segment FC is congruent to line segment ED and the line segment EF is congruent to line segment CD. The resulting figure EFCD is a parallelogram. Since each four lines are perpendicular to each other, then the parallelogram is a rectangle.)

           

            

              

         
If you will substitute -ϕ to the above equation, then

How about for Tan (θ + ϕ)?

Ok, let's do for Tan (θ + ϕ). We don't have to use the figure again to derive Tan (θ + ϕ).  We know that 

            
If we divide both the numerator and denominator by cos ϕ cos θ, then the above equation will be

         
If you will substitute -ϕ to the above equation, then