NIMCET 2020 — Mathematics PYQ

Concept:

Trigonometric Identities:
- $\cos (A\pm B)=\cos A\cos B\mp \sin A\sin B$.
- $2\cos A\cos B=\cos (A-B)+\cos (A+B)$.
- $2\sin A\sin B=\cos (A-B)-\cos (A+B)$.

Trigonometric Values:
- $\cos 60^{\circ }=\frac{1}{2}$.
- $\cos 36^{\circ }=\frac{\sqrt{5}+1}{4}$

Trigonometric Ratios for Allied Angles:
$\sin (-\theta )=-\sin \theta .$
$\cos (-\theta )=\cos \theta .$
$\sin (\pi +\theta )=(-1{)}^n\sin \theta .$
$\cos (\pi +\theta )=(-1{)}^n\cos \theta .$
$\sin \left[(2n+1)\frac{\pi }{2}+\theta \right]=(-1{)}^n\sin \theta .$
$\cos \left[(2n+1)\frac{\pi }{2}+\theta \right]=(-1{)}^n(-\sin \theta ).$

Calculation:
$\sin 12^{\circ }\sin 48^{\circ }\sin 54^{\circ }$
$=\frac{2\sin 48^{\circ }\sin 12^{\circ }}{2}\sin 54^{\circ }$
$=\left[\frac{\cos (48^{\circ }-12^{\circ })-\cos (48^{\circ }+12^{\circ })}{2}\right]\sin (90^{\circ }-36^{\circ })$
$=\frac{\cos 36^{\circ }-\cos 60^{\circ }}{2}\sin 36^{\circ }$
$=\left(\frac{1}{2}\right)\left(\frac{\sqrt{5}+1}{4}-\frac{1}{2}\right)\left(\frac{\sqrt{5}+1}{4}\right)$
$=\left(\frac{1}{2}\right)\left(\frac{\sqrt{5}-1}{4}\right)\left(\frac{\sqrt{5}+1}{4}\right)$
$=\left(\frac{1}{2}\right)\left(\frac{5-1}{16}\right)$

$=\frac{1}{8}.$