JEE Advanced 2021 — Paper 1 — Question 30

$\mathrm{AB}=\sqrt{23}=\mathrm{C}$ ,$B C=3=a$ ,$C A=4=b$

$\frac{\cot A+\cot C}{\cot B} ; \frac{\frac{\cos A}{\sin A}+\frac{\cos C}{\sin C}}{\frac{\cos B}{\sin B}}$

$=\frac{\cos A \sin C+\cos C \sin A}{\sin A \cdot \sin C \cdot \frac{\cos B}{\sin B}}=\frac{\sin (A+C) \cdot \sin B}{\sin A \cdot \sin C \cdot \cos B}=\frac{\sin B \cdot \sin B}{\sin A \cdot \sin C \cdot \cos B}$

$=\frac{b^{2}}{a c \cdot \frac{\left(a^{2}+c^{2}-b^{2}\right)}{2 a c}}=\frac{2 b^{2}}{a^{2}+c^{2}-b^{2}}=\frac{2 \times 16}{9+23-16}=\frac{2 \times 16}{32-16}=\frac{2 \times 16}{16}=2$