Deriving $\cosh \frac{x}{R} = \frac{1}{\sqrt{1- u^2 - v^2}}$ from $x = \frac{R}{2}\;\log\left(\frac{1+\sqrt{u^2 + v^2}}{1-\sqrt{u^2 + v^2}}\right)$

Question

Let $$x = \frac{R}{2}\;\log\left(\frac{1 + \sqrt{u^2 + v^2}}{1 - \sqrt{u^2 + v^2}}\right)$$ from which I derived that $$\tanh \frac{x}{R} = \sqrt{u^2 + v^2}$$ I have difficulty somehow in deriving the formula $$\cosh \frac{x}{R} = \frac{1}{\sqrt{1- u^2 - v^2}}$$ Any hints?