If tan &#x2061;<!-- ⁡ --> ( &#x03C0;<!-- π --> 12 </mfrac

Cooper Doyle

Cooper Doyle

Answered question

2022-07-01

If
tan ( π 12 x ) , tan ( π 12 ) , tan ( π 12 + x )
in order are three consecutive terms of a GP then what is sum of all possible values of x.
I am not getting any start, can anybody provide me a hint?

Answer & Explanation

owerswogsnz

owerswogsnz

Beginner2022-07-02Added 12 answers

Use :
(1) ( tan π 2 ) 2 = tan ( π 12 x ) tan ( π 12 + x )
(Condition for G.P)
And :
tan ( A ± B ) = tan A ± tan B 1 tan A tan B
Let tan π 12 = c
Put in (1)
c 2 = c + tan x 1 c tan x c tan x 1 + c tan x c 2 c 4 tan 2 x = c 2 tan 2 x tan x = 0
Sum of solutions in a bounded interval will be sum all angles in the form k π   ;   k Z

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