For given random values $$X_i \sim\mathcal{N}(0,1)$$ and $$\frac{X_i-\mu}{\sigma}=\tilde{X_i}\sim\mathcal{N}(\mu,\sigma),\,\mu\in\mathbb{R},\,\sigma>0$$ prove
$$\left(\frac{\left({\frac{{X_j}-\mu}{\sigma}}-{\frac{{X_k}-\mu}{\sigma}}\right)^2}{\frac{1}{n}\sum_{j=1}^n\left({\frac{X_j-\mu}{\sigma}}-\left(\frac{1}{n}\sum_{j=1}^n{\frac{X_j-\mu}{\sigma}}\right)\right)^2}\right)$$ $\overset{\text{d}}{=}$ $$\left(\frac{\left({X_j-X_k}\right)^2}{\frac{1}{n}\sum_{j=1}^n\left({X_j}-\left(\frac{1}{n}\sum_{j=1}^n{X_j}\right)\right)^2}\right)$$ So I have to eliminate $\mu$ and $\sigma$ from the equation above. I dont know how to proceed.
Just factor out $1 / \sigma$. In the numerator and in the denomitator $1 / \sigma$ are in squared brackets. If you factor out them, the factor is in both cases $1 / \sigma ^2$. The factors can be cancelled. And the numerator becomes $X_j-\mu-(X_k-\mu)$. This gives $X_j-X_k$