NIMCET 2016 — Mathematics PYQ

Concept:
$cosA+cosB=2cos\frac{A+B}{2}cos\frac{A-B}{2}$
$cosA-cosB=-2sin\frac{A+B}{2}sin\frac{A-B}{2}$
$sinA+sinB=2sin\frac{A+B}{2}cos\frac{A-B}{2}$
$sinA-sinB=2cos\frac{A+B}{2}sin\frac{A-B}{2}$

Calculation

$$\cos 20^{\circ }+\cos 100^{\circ }+\cos 140^{\circ }=2\cos \frac{20+100}{2}\cos \frac{100-20}{2}+\cos 140^{\circ }$$
$$=2\cos 60^{\circ }\cos 40^{\circ }+\cos 140^{\circ }$$
$$=2\times \frac{1}{2}\cos 40^{\circ }+\cos 140^{\circ }$$
$$=\cos 40^{\circ }+\cos 140^{\circ }$$
$$=2\cos \frac{140+40}{2}\cos \frac{140-40}{2}$$
$$=2\cos 90^{\circ }\cos 50^{\circ }$$
$$=0(\because \cos 90^{\circ }=0)$$
$$\therefore \cos 20^{\circ }+\cos 100^{\circ }+\cos 140^{\circ }=0$$