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)#.