Skip to main content

how is tan (A+B) = [tan A + tan B]/[1 - tan A tan B]?

How is tan (A+B) = [tan A + tan B]/
[1 - tan A tan B]?

Method 1:
Let A = 30 deg and B = 45 deg.
LHS = tan (30+45) = tan 75 = 3.732050808
RHS = [tan A + tan B]/[1 - tan A tan B]
= [tan 30 + tan 45]/[1 - tan 30 tan 45]
= [0.577350269 + 1]/[1 - 0.577350269*1]
= 1.577350269/0.42264973
= 3.732050808 = LHS
Proved.
Method 2:
tan (A+B) = [tan A + tan B]/[1 - tan A tan B]
RHS = [tan A + tan B]/[1 - tan A tan B]
=[(sin A/cos A) + (sin B/cos B)]/[1-(sin A/cos A)(sin B/cos B)
= [sin A cos B + cos A sin B]/[cos A cos B][1 - sin A sin B/(cos A cos B)]
= sin (A+B)/{[cos A cos B][cos A cos B - sin A sin B]/(cos A cos B)}
= sin (A+B)/[cos A cos B - sin A sin B
= sin (A+B)/cos (A+B)
= tan (A+B) = LHS.
Proved.
Thanks.
source:Quara

Comments

Popular posts from this blog

There are five people in a room,i come and killed 4 how many remains?

The answer is four. I can quite confidently say that based on the wording given, the answer is four. First of all, we need to establish some things. I am assuming: All five people were alive before entering The four people that were killed are the only people that have died I can figure this out because of the last word: “remains.” “Remains” can be one of two things: a noun or a verb. As a verb , there are multiple answers to the question. First, if there are five people and four are killed, one person is left alive. However, you are in there as well, so would that be two remaining? Or should we count everyone in the room, which would be six people? Well, none of this matters. This is because the question states “How many remains?” As a verb, “remains” applies to only a few certain subjects: he/she/one (he remains/she remains/one remains). Otherwise it would be I remain/You remain/They remain/We remain. If “remain” was to be used as a verb, it would’ve said “How man

What are some mind-blowing facts about mathematics?

Prime Generating Polynomials : The polynomial,  n 2 + n + 41 n 2 + n + 41  can be used to produce 40 primes for consecutive integer values 0≤n≤39.  This property was discovered by Euler . Similarly, the incredible formula,  n 2 − 79 n + 1601 n 2 − 79 n + 1601  was discovered, which produces 80 primes for the consecutive values 0≤n≤79! Kaprekar’s Constant : 6174 is known as Kaprekar’s Constant , after the Indian Mathematician D.R. Kaprekar. Take any four-digit number, using at least two different digits (Leading zeros are allowed.) Arrange the digits in descending and then in ascending order, to get two four-digit numbers, adding leading zeroes if necessary. Subtract the smaller number from the larger number and go back to step 2. The above process will always reach the fixed number, 6174, taking at most 7 iterations. Try it yourself! ( Note that after the first subtraction or the subsequent subtractions, the result obtained is always a multiple of 9!) Collatz Con