If $a+b$ is an irrational number, is $a-b$ an irrational number, too?

Question

Question 1: If $a+b$ is an irrational number. Is $a-b$ an irrational number, too?

Question 2: If $\cos(a)-\sin(a)$ is irrational, Is $\sin(a)-\cos(a)$ irrational, too?

Answer 1

HINT: Try for an example with $a=b$.

For the second question, note that $x-y=-(y-x)$.

Answer 2

No on 1. $ (2+\sqrt{5}) + (1 + \sqrt{5}) = 3 + 2\sqrt{5}$, while $ (2+\sqrt{5}) - (1 + \sqrt{5}) = 1$.

Answer 3

Hard problem:

Is it possible to pick $a$ and $b$ such that $a+b$ is irrational but $a-b$ is rational?

Easy problem:

Is it possible to pick irrational $c$ and rational $d$ such that you can find $a,b$ such that $a+b = c$ and $a-b = d$?

They're really the same problem, of course....