If, the resultant of two forces (A + B) and (A - B) is $\sqrt{{\mathrm{A}}^2+{\mathrm{B}}^2}$, then the angle between these forces is 

Here, $$\begin{gathered} \mathrm{P}=(\mathrm{A}+\mathrm{B}) \\ \mathrm{Q}=(\mathrm{A}-\mathrm{B}) \end{gathered}$$

and $R=\sqrt{A^2+B^2}$
$$\begin{gathered} \Rightarrow \mathrm{cos\theta }=\frac{{\mathrm{R}}^2-{\mathrm{P}}^2-{\mathrm{Q}}^2}{2\mathrm{PQ}} \\ =\frac{\left({\mathrm{A}}^2+{\mathrm{B}}^2\right)-(\mathrm{A}+\mathrm{B}{)}^2-(\mathrm{A}-\mathrm{B}{)}^2}{2(\mathrm{A}+\mathrm{B})(\mathrm{A}-\mathrm{B})}=-\left[\frac{{\mathrm{A}}^2+{\mathrm{B}}^2}{2\left({\mathrm{A}}^2-{\mathrm{B}}^2\right)}\right] \\ \mathrm{\theta }={\cos }^{-1}\left[-\frac{{\mathrm{A}}^2+{\mathrm{B}}^2}{2\left({\mathrm{A}}^2-{\mathrm{B}}^2\right)}\right] \end{gathered}$$