Question 1208813
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Solve for x:  (sin x + cos x)/(1 - tan x) = 0
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        The solution in the post by Edwin is incomplete.

        I came to bring a complete solution.



<pre>
The domain (the set of real numbers, where left side of the equation
is defined) is  tan(x) =/= 1.

So, the prohibited values of x are  {{{pi/4 + k*pi}}},  k = 0, _/-1, +/-2, . . . 


We are looking for solutions of the given equation that are in its domain.


In the domain, the given equation is equivalent to

    sin(x) + cos(x) = 0,    (1)

or

    sin(x) = -cos(x).       (2)


The solutions to this equation can not be with cos(x) = 0 (since then sin(x) = 1, 
and this equation is not held) .


Therefore, we can divide both sides of equation (2) by cos(x).  We get then

    tan(x) = -1.


The solutions to this equation are

    x = {{{(3/4)*pi + k*pi}}},  or  135° + 180°*k,  k = 0, +/-1, +/-2, . . . 


<U>ANSWER</U>.  The solutions to the given equation are   x = {{{(3/4)*pi + k*pi}}},  or  135° + 180°*k,  k = 0, +/-1, +/-2, . . . 
</pre>

Solved.