Because if you look at the graph of y=x and y = cos(x), you'll see that they touch at some point.
Graph both equations on the same plane, then look at the square area marked by the points (0,0), (0,pi/2), (pi/2,0) and (pi/2, pi/2). The graph of y=x goes straight from the (0,0) corner on the lower left to the (pi/2, pi/2) corner on the upper right. This splits the square along the diagonal. The graph of y=cos(x) enters this box on the left edge at (0,1) and leaves it on the right edge at the corner (pi/2, 0), so at some point it has to cross the y=x line.
