How do you solve #Tan x + cot x -2 =0 #?

1 Answer
Burglar
Jun 18, 2016

The solution is #x=\pi/4+npi#.

Explanation:

#cot(x)=1/tan(x)# then

#tan(x)+cot(x)-2=0#

#tan(x)+1/tan(x)-2=0#

#(tan(x)^2+1-2tan(x))/tan(x)=0#

#tan(x)^2+1-2tan(x)=0#

this is a square

#tan(x)^2+1-2tan(x)=0#

#(tan(x)-1)^2=0#

doing the square root we obtain

#tan(x)-1=0#

#tan(x)=1#

#x=arctan(1)=\pi/4#.

This is called the principal solution. We know that the tangent is periodic of period #pi# then also #\pi/4+pi#, #\pi/4+2pi#, #\pi/4+3pi#, etc. are solutions.
The generic solution is then #pi/4+npi# where #n# is any positive or negative integer number.

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