# Cotangent of the Difference Between Two Angles

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(9 ^{2} + 19^{2})**

**= sqrt(442)**and get

**sin (A – B) = ± 19/ sqrt(442)**and

**cos (A – B) = ± 9/ 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.