What is x if 2*sin x + 1 = tan x + 2*sin x * tan x,  if x lies in [0; 2*pi] ?

Expert Answers

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We have to find x so that 2*sin x + 1 = tan x + 2*sin x * tan x, where x is in the interval [ 0 ; 2pi]

2*sin x + 1 = tan x + 2*sin x * tan x

=> 2*sin x - 2*sin x * tan x = tan x - 1

=> 2*sin x( 1 - tan x) = -( 1- tan x)

=> 2*sin x( 1 - tan x) + ( 1 - tan x) = 0

=> (1 - tan x)( 2*sin x + 1) = 0

=> tan x = 1 and 2*sin x = -1

=> tan x = 1 and sin x = -1/2

=> x = arc tan 1 and x = arc sin (-1/2)

=> x = 45 degree and 330 degree.

Therefore x= 45 degree and 330 degree.

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