To solve this problem we must apply the cosine formula for sum of angles, which is:
cos(A ± B) = cos A . cos B ± sin A . sin B
For this case we may write:
cos(A - B) = cos A . cos B + sin A . sin B
Now let's plug the given information in the equation above. We get:
cos(A - B) = 12/13 . cos B + sin A . 21/29 (*)
Now we must find the unknown values. Apply fundamental trigonometric relation:
sin ² A + cos ² A = 1 (I)
and
sin ² B + cos ² B = 1 (II)
If cos A = 12/13 then equation (I) becomes:
sin² A + (12/13)² = 1
sin² A = 1 - (12/13)²
sin² A = 1 - 144/169
sin² A = 169/169 - 144/169
sin² A = 25/169
Since angle A is a positive angle, we get:
sin A = 5/13
Now, let's do the same thing for equation (II). If sin B = 21/29 , we have:
(21/29)² + cos² B = 1
cos² B = 1 - (21/29)²
cos² B = 1 - 441/841
cos² B = 841/841 - 441/841
cos² B = 400/841
Since B is positive angle, we get:
cos B = 20/29
Now we can go back the original equation (*). We get:
cos(A - B) = 12/13 . cos B + sin A . 21/29
cos(A - B) = 12/13 . 20/29 + 5/13 . 21/29
cos(A - B) = 240/377 + 105/377
cos(A - B) = 345/377
Answer: cos(A - B) = 345/377