How can you solve this equation for all values of x in the domain [0, 2pi) with exact values?

Question

cot(x) - tan (x) = $2sqrt(3)$

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Answer 1 · 2016-09-06T05:14:22

The Soln. Set is ${pi/12, 13pi/12} sub [0,2pi)$ .

$cotx-tanx=2sqrt3$

$:. 1/tanx-tanx=2sqrt3$

$:. (1-tan^2x)/tanx=2sqrt3, i.e., (1-tan^2x)/(2tanx)=sqrt3$

Recalling that, $tan2x=(2tanx)/(1-tan^2x)$ , we have,

$tan2x=1/sqrt3=tan(pi/6)$

$;. 2x=pi/6, or, x=pi/12" is one soln."$

As the Principal Period of $tan$ fun. is $pi$ , the other soln. is

$pi+pi/12=13pi/12$ .

Thus, the Soln. Set is ${pi/12, 13pi/12} sub [0,2pi)$ .