Question 1078914
.
 tgx + ctgx = -2
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Let y = tg(x) be the new variable.


Then the given equation takes the form  

{{{y + 1/y}}} = -2.


Multiply by y both sides. You will get

{{{y^2 + 1}}} = -2y,   or

{{{y^2 + 2y + 1}}} = 0,

{{{(y+1)^2}}} = 0  --->  y+1 = 0.


Hence, tg(x) + 1 = 0  --->  tg(x) = -1  --->  x = {{{3pi/4}}}  and/or  x = {{{7pi/4}}}.

<U>Answer</U>.  There are 2 solutions:  x = {{{3pi/4}}}  and/or  x = {{{7pi/4}}}.
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