cos(2x)cos(x) + sin(2x)sin(x) = 1

You are going about it wrong.  You were
expected to recognize that the left side 
of that equation looks very much 
like the right side of the double
angle identity:

cos(A-B) = cos(A)cos(B) + sin(A)sin(B)

where A = 2x and B = x

So the left side of your original equation
can be replaced by cos(2x-x) from the left
side of the identity.

and immediately you can write

  cos(2x-x) = 1

     cos(x) = 1

So the answer is 

          x = 0 

Or, if your teacher wants all real solutions it is 

          x = 2np for all integers n.

Edwin