Calculate without using tables:
$\dfrac{96 \sin 80^\circ \sin 65^\circ \sin 35^\circ}{\sin 20^\circ + \sin 50^\circ + \sin 110^\circ}$
The given expression is: $\;\;$ $\dfrac{96 \sin 80^\circ \sin 65^\circ \sin 35^\circ}{\sin 20^\circ + \sin 50^\circ + \sin 110^\circ}$ $\;\;\; \cdots \; (1)$
Denominator of the given expression is
$\sin 20^\circ + \sin 50^\circ + \sin 110^\circ$
$= \sin \left(2 \times 10^\circ\right) + \left(\sin 50^\circ + \sin 110^\circ\right)$
$= 2 \sin 10^\circ \cos 10^\circ + 2 \sin \left(\dfrac{110^\circ + 50^\circ}{2}\right) \cos \left(\dfrac{110^\circ - 50^\circ}{2}\right)$
$= 2 \sin 10^\circ \cos 10^\circ + 2 \sin 80^\circ \cos 30^\circ$
$= 2 \sin 10^\circ \cos 10^\circ + 2 \sin \left(90^\circ - 10^\circ\right) \cos \left(90^\circ - 60^\circ\right)$
$= 2 \sin 10^\circ \cos 10^\circ + 2 \cos 10^\circ \sin 60^\circ$
$= 2 \cos 10^\circ \left(\sin 60^\circ + \sin 10^\circ\right)$
$= 2 \cos \left(90^\circ - 80^\circ\right) \times 2 \sin \left(\dfrac{60^\circ + 10^\circ}{2}\right) \cos \left(\dfrac{60^\circ - 10^\circ}{2}\right)$
$= 4 \sin 80^\circ \sin 35^\circ \cos 25^\circ$
$= 4 \sin 80^\circ \sin 35^\circ \cos \left(90^\circ - 65^\circ\right)$
$= 4 \sin 80^\circ \sin 35^\circ \sin 65^\circ$ $\;\;\; \cdots \; (2)$
In view of $(2)$, the given expression $(1)$ becomes
$\dfrac{96 \sin 80^\circ \sin 65^\circ \sin 35^\circ}{4 \sin 80^\circ \sin 35^\circ \sin 65^\circ}$
$= \dfrac{96}{4} = 24$