$1+\cos 10^{\circ}+\cos 20^{\circ}+\cos 30^{\circ}=$

$1+\cos 10^{\circ}+\cos 20^{\circ}+\cos 30^{\circ}=$
  1. $4 \sin 10^{\circ} \sin 20^{\circ} \sin 30^{\circ}$
  2. $4 \cos 5^{\circ} \cos 10^{\circ} \cos 15^{\circ}$
  3. $4 \cos 10^{\circ} \cos 20^{\circ} \cos 30^{\circ}$
  4. $4 \sin 5^{\circ} \sin 10^{\circ} \sin 15^{\circ}$

Solution

We have, $ \begin{aligned} & 1+\cos 10^{\circ}+\cos 20^{\circ}+\cos 30^{\circ} \\ & \quad=\left(1+\cos 10^{\circ}\right)+\left(\cos 20^{\circ}+\cos 30^{\circ}\right) \\ & =2 \cos 5^{\circ}+2 \cos 25^{\circ} \cos 5^{\circ} \\ & =2 \cos 5^{\circ}\left(\cos 5^{\circ}+\cos 25^{\circ}\right) \\ & =2 \cos 25^{\circ}\left(2 \cos 15^{\circ} \cos 10^{\circ}\right) \\ & =4 \cos 5^{\circ} \cos 10^{\circ} \cos 15^{\circ} \end{aligned} $

Asked in: AP EAMCET 2017 (26 Apr Shift 1)

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