I saw this problem on Quora today. So far, two answers have been submitted. I included the first solution here as the “conventional solution”. The second is very much like the first solution in that both used the addition formula tan (A – B) = (tan A – tan B)/(1 + tan A tan B). There are other formulas we have to memorize if we want to know the values of the other trigonometric ratios of a sum or difference of two angles.
In Vedic Mathematics, we can express an angle as a Pythagorean triple. And angle A between the base and hypotenuse of a right triangle can be expressed as A) base, height, hypotenuse. It can be A) adjacent, opposite, hypotenuse in conventional trigonometry, or simply A) x, y, r in analytic geometry.
Thus, in our problem, since tan A = 3/5, we can describe angle A as A) 5, 3, – . Note that since the value of the hypotenuse will not be involved in the calculations, we do not have to write it down. Similarly, since tan B = – 2/3 we can have B) 3, – 2, – or B) – 3, 2, –.
By representing angles as triples, we do not have to memorize addition, subtraction or double angle formulas. All we need to know is triple addition which is a form of Vertically and Crosswise.
In general if
A) x, y, r
B) X, Y, R
A+ B) xX – yY, xY + Xy, rR
For subtraction, we just have to reverse the sign of the height of the second angle. Thus
A – B) xX + yY, Xy – xY, rR
After finding A -B) 9, 19 – in the problem given, we can immediately say that cot (A – B) = 9/19.
Now if we use B) -3, 2 –, we will have
A) 5, 3, –
B) – 3 , 2, –
A – B) – 9, – 19
Thus we will also have cot (A – B) = 9/19.
If we want to know sin (A – B) or cos (A – B), we just have to compute r = sqrt(92 + 192) = sqrt(442) and get sin (A – B) = ± 19/ sqrt(442) and cos (A – B) = ± 19/ sqrt(442). The correct sign can be determined if we were given in which quadrants the angles A and B are located.
Note that triples were discussed by Sir Benmar Mariano in his Fresh Approach to Trigonometry talk in the Inspirational Maths from India (Year 4) webinars held earlier this month. You can still view the recording of all presentations in IMI 4 until the end of 2022 if you register before Dec 31, 2021. Send a message to MATH-Inic Philippines at Facebook for details.