Prove this: tan(20)*tan(40)*tan(100)= -√3?

Question

Thank you

Answer 1

hi

triple angle
tan(3a) = ( 3 tan(a) - tan^3(a))/(1 - 3 tan^2(a) )

notable angle
tan(30) = V3/3

tan(30) = ( 3 tan(10) - tan^3(10))/(1 - 3 tan^2(10) )

tan(20)*tan(40)
= tan(30-10)*tan(30+10)
Sum/difference angle
= (tan^2(30) - tan^2(10)) / ( 1 - tan^2(30)tan^2(10))
= ( 1/3 - tan^2(10)) / ( 1 - 1/3 tan^2(10))
= (1 - 3 tan^2(10)) / ( 3 - tan^2(10))

tan(100) = - 1/tan(10)

finally

tan(20)*tan(40)*tan(100) =
= - (1 - 3 tan^2(10)) / ( 3 tan(10) - tan^3(10))
= -1 / tan(30)
= - V3

bye

Answer 2

A tough problem! I finally found a method that works.
Here is the solution:
First tan 100 = -tan 80 and √3 = tan 60
So our problem becomes:
Prove that tan 20 tan 40 tan 80 = tan 60.
I will show that both sides reduce to the same quantity,
which will prove the result.
Let's recall the identities
tan(x+y) = (tan x + tan y)/(1-tan x tan y)
tan(x-y) = (tan x - tan y)/( 1+tan x tan y).
So, using tan 60 = √3, and substituting, we get
tan 40 = tan(60-20) = (√3 - tan 20)/(1 + √3 tan 20)
tan 80 = tan(60+20) = (√3 +tan 20)/(1 - √3 tan 20).
Multiplying everything together, we get
tan 20 tan 40 tan 80 = (3 tan 20--tan³ 20)/(1 - 3 tan² 20).

Now let's look at tan 60 = tan(20 + 20 + 20)
= (tan 40 + tan 20)/(1-tan 40 tan 20)
= (2 tan 20/ (1-tan² 20) + tan 20)
------------------------------------------------- .
1 - tan 20(2 tan 20/(1 - tan² 20) )

But this simplifies to
(3 tan 20 -tan³ 20)/ (1-3 tan² 20)
Since both sides reduce to the same
quantity, we are done!

Answer 3

there is not any might desire to get slowed down in convoluted arguments right here because of fact there is an elephant interior the room which all and sundry seems to be ignoring. permit me please draw interest to the character of this elephant. If God used Jesus Christ (and the Holy Spirit, permit's not forget!) to create all issues, then Jesus Christ can't be a created element Himself. he's the two the writer as is the daddy and the Holy Spirit. No introduction became refrained from all 3 persons interacting in subliminal solidarity and cohesion of purpose. merely inspect John a million verse 3 - "All issues have been made by using him; and with out him became not something made that became made." for this reason, Jesus, the notice, won't be able to have been made - he can't be a creature! Simples! Oh, by using the way, i does not set plenty shop by using Arius and his ideals, in view that he suggested, "the daddy keeps to be ineffable to the Son, and the notice can neither see nor understand the daddy completely and accurately... Fourthly, the Son might desire to be in hazard of replace or perhaps sin." His perspectives have been based in Origen and went back to Platonic roots. (Early Christian Doctrine by using Canon J N D Kelly, p 228 4th ed. Adam & Charles Black, London)

Answer 4

I'm pretty sure you'll have to use the identity
cos20*cos40*cos80 = 1/8
for that one...