We know that root 2 is an irrational number
So it is enough to prove that 1/root 2 is irrational.
If possilble assume that 1/root 2 is rational.
so there exists an integers p and q such that 1/root 2 = p/q
since 1/root 2 not equal to zero, so p is not equal to zero.
so 1/(1/root 2) exists.
root2 = 1/(1/root 2) = 1/(p/q) = q/p
since p and q are integers root 2 can be written as the ratio of those two integers.
this implies root 2 is rational. which is a contradiction. So 1/root 2 is irratonal.
(i)we know that √2 is irrational.
(ii)assume that 1/√2 is rational.
(iii)therefore,1/√2=p/q,(q is not equal to 0) and p and q are co-prime integers.
(iv)on cross multiplying
(v)that is √2 is rational which is a contradition.
   therefore,1/√2 is irrational.