SOLUTION: Assume 3 cos(x) − 4 sin(x) = 4 and 4 cos(x) + 3 sin(x )= 3 Find the exact (numeric) value of cot(x).

3 cos(x) - 4 sin(x) = 4 

Divide through by sin(x)

3 - 4 = 

3cot(x) - 4 = 4csc(x)

-------------------

4 cos(x) + 3 sin(x )= 3

Also divide through by sin(x)

4 + 3 = 

4cot(x) + 3 = 3csc(x)

So we have the system:

3cot(x) - 4 = 4csc(x)
4cot(x) + 3 = 3csc(x)

To eliminate the cosecant terms, multiply the first through 
by -3 and the second through by 4, and add them term by term:

-9cot(x) + 12 = -12csc(x)
16cot(x) + 12 =  12csc(x)
-------------------------
 7cot(x) + 24 = 0

      7cot(x) = 24
       cot(x) = 

Edwin