cos(42+x) + sin(-x+48) = 0

  is equivalent to

cos(42+x) = -sin(-x+48)

  is equivalent to

cos(42+x) = sin(x-48)

  is equivalent to 

cos(42+x) = cos(90-(x-48))

  is equivalent to 

cos(42+x) = cos(138-x).


The last equation implies that  EITHER  42 + x = 138 - x  OR  42 + x = 138 - x + 360.

Let us consider both opportunities one after another.


1.  42 + x = 138 - x  ===>  2x = 138 - 42 = 96  ===>  x = 48.

    It is one solution. It satisfies the original equation, as you can easily check.


2.  42 + x = 138 - x  + 360  ===>  2x = 138 - 42 + 360  ===>  x = 48 + 180.

    It is another solution. It also satisfies the original equation, as you can easily check.


Answer.  The solutions are  x = 48  and  x = 48+180 = 228.