3*cosx - 4*sinx = 5.            (1)

Let us rewrite this equation in the form

.cos(x) - .sin(x) = 1.     (2)

Now consider the right-angled triangle with the sides (3,4,5), and let  be its smallest of two acute angles.

Then  =  and  = ,

so we can rewrite the left side of the equation (2) in the form 

 = 1,   or

 = 1.

It implies that 

 =    and

 = 0.                  (3)

But, from the other side, 

 =  + .

Now substitute here  =  and  = , and you will get

 +  = .   (4)

As a last step, multiply both sides of (4) by 5, and you will get

4*cos(x) + 3*sin(x) = 0.

The problem is solved.

Answer.  If 3cosx - 4sinx = 5, then 3sinx + 4cosx = 0.