Given $\mathrm{a}=3, \mathrm{~b}=7, \mathrm{c}=8$
Now, $\sin \left(\frac{\mathrm{B}}{2}\right) \tan \left(\frac{\mathrm{C}-\mathrm{A}}{2}\right)=\sin \frac{\mathrm{B}}{2}\left(\frac{\mathrm{c}-\mathrm{a}}{\mathrm{c}+\mathrm{a}}\right) \cot \frac{\mathrm{B}}{2}$ \{By Law of tangent\}


Take, $\cos ^2 \frac{\mathrm{B}}{2}=\frac{1+\cos \mathrm{B}}{2}$
So, $\cos \mathrm{B}=\frac{\mathrm{a}^2+\mathrm{c}^2-\mathrm{b}^2}{2 \mathrm{ac}}=\frac{9+64-49}{2 \times 3 \times 8}=\frac{24}{6 \times 8}=\frac{1}{2}$
Put value of $\cos B$ in the value $\cos ^2\left(\frac{B}{2}\right)$
$$
\begin{aligned}
& \cos ^2 \frac{B}{2}=\frac{1+\frac{1}{2}}{2}=\frac{3}{4} \\
& \cos \left(\frac{B}{2}\right)=\frac{\sqrt{3}}{2}
\end{aligned}
$$
From equation (i),
$$
\sin \frac{\mathrm{B}}{2} \tan \frac{\mathrm{C}-\mathrm{A}}{2}=\frac{5}{11} \times \frac{\sqrt{3}}{2}=\frac{5 \sqrt{3}}{22}
$$
So, option (d) is correct.