If the segment joining the points $A\left(a,b\right)andB\left(c,d\right)$  subtends an angle $\theta$  at the origin, then

Let O be the origin. Then $O{A}^2=a^2+b^2,O{B}^2=c^2+d^2$  and $A{B}^2={\left(c-a\right)}^2+{\left(d-b\right)}^2$ 

Using cosine formula in $\Delta OAB,$  we have

$A{B}^2=O{A}^2+O{B}^2-2.OA.OB\cos \theta$

$\Rightarrow$   $\left(c-a^2\right)+{\left(d-b\right)}^2=a^2+b^2+c^2+d^2-2\sqrt{a^2+b^2}\sqrt{c^2+d^2}\cos \theta$

$\Rightarrow$ $c^2+a^2-2ac+d^2+b^2-2bd=a^2+b^2+c^2+d^2-2\sqrt{a^2+b^2}\sqrt{c^2+d^2}\cos \theta$

$\Rightarrow$ $2\left(ac+bd\right)=2\sqrt{a^2+b^2}\sqrt{c^2+d^2}\cos \theta$

$\Rightarrow$   $\cos \theta =\frac{ac+bd}{\sqrt{a^2+b^2}\sqrt{c^2+d^2}}.$