tan(A) = 4/3 gets you an angle of 53.13010235 using a calculator.
tan(B) = 12/5 gets you an angle of 67.38013505 using a calculator.
Since pi/2 is the same as 180 degrees, then the requirement that 0 < A < B < pi/2 is satisfied.
A-B = -14.2500327 degrees
cos(-14.2500327) = .9692307692 using your calculator.
If you use the cos(A-B) formula, you should get the same answer.
That formula is cos(A-B) = cos(A)*cos(B) + sin(A)*sin(B)
Since you have A and B, it's just a matter of plugging the values into the formula.
You will get cos(A-B) = .969230769 using that formula.
Once you know the angles, the formula is unnecessary, but this just goes to show you will get the same answer either way.
Below are the calculations using the cosine(a-b) formula.
tan(a) = 4/3
tan(b) = 12/5
make 2 right triangles, one with angle a and one with angle b.
use those triangle to solve for the missing side.
the picture of those triangles is shown below.
using the pythagorean formula to solve for the missing sides yields.
sin(a) = 4/5
cos(a) = 3/5
sin(b) = 12/13
cos(b) = 5/13
now use the cos(a-b) formula to find your answer.
cos(a-b) = cos(a)cos(b) + sin(a)sin(b)
this becomes:
cos(a-b) = 3/5 * 5/13 + 4/5 * 12/13 which becomes:
15/65 + 48/65 which becomes:
63/65
the decimal equivalent of this is .9692307692 which is the same value derived earlier before I knew the correct way to solve this using the sum and difference identity formulas.
