$ = a\sqrt {{{\sin }^2}\alpha + {{\cos }^2}\alpha + {{\cos }^2}\beta + {{\sin }^2}\beta - 2\cos \alpha \cos \beta - 2\sin \alpha \sin \beta } $
$ = a\sqrt {2\left\{ {1 - \cos \,(\alpha - \beta )} \right\}} $
$= 2a\,\sin \,\left( {\frac{{\alpha - \beta }}{2}} \right)$
Trick : Put $a = 1,\,\,\alpha = \frac{\pi }{2},\,\beta = \frac{\pi }{6},$ then the points will be $(0, 1)$ and $\left( {\frac{{\sqrt 3 }}{2},\,\,\frac{1}{2}} \right)$.
Obviously, the distance between these two points is 1 which is given by $(d)$.
$\left\{ {\,\,2a\,\sin \frac{{\alpha - \beta }}{2} = 2 \times 1 \times \sin \frac{{(\pi /2) - (\pi /6)}}{2} = 2 \times \frac{1}{2} = 1} \right\}$
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Statement $-1$ :Variance of $2{x_1}\;,2\;{x_2}\;,\;.\;.\;.\;,2{x_n}$ is $4{\sigma ^2}$ .
Statement $-2$: Arithmetic mean $2{x_1}\;,2\;{x_2}\;,\;.\;.\;.\;,2{x_n}$ is $4\bar x$.