|cos(z)| is bounded by 1 for all z in the complex plane, where z is the complex number. True or False? Justify your answer.?

This is false.

To see this, observe that by writing z = x + iy (with x, y real):
cos z = cos(x + iy)
.........= cos x cos(iy) - sin x sin(iy)
.........= cos x cosh y - i sin x sinh y.

Therefore,
|cos z| = √[cos²x cosh²y + sin²x sinh²y]
............= √[cos²x (1 + sinh²y) + (1 - cos²x) sinh²y]
............= √(cos²x + sinh²y).

As y→∞, we see that |cos z|→∞. Thus, |cos z| is unbounded.

I hope this helps!