35^ + 85^ + 155^ = - Vidyadip

MCQ

$\cos35^\circ+\cos85^\circ+\cos155^\circ=$

✓
$0$
B
$\frac{1}{\sqrt3}$
C
$\frac{1}{\sqrt2}$
D
$\cos275^\circ$

✓

Answer

Correct option: A.

$0$

$\cos35^\circ+\cos85^\circ+\cos155^\circ$
$=\ 2\cos\Big(\frac{35^\circ+85^\circ}{2}\Big)\cos\Big(\frac{35^\circ-85^\circ}{2}\Big)+\cos155^\circ$ $\Big[\because\ \cos\text{A}+\cos\text{B}=2\cos\Big(\frac{\text{A+B}}{2}\Big)\cos\Big(\frac{\text{A}-\text{B}}{2}\Big)\Big]$
$=\ 2\cos60^\circ\cos(-25^\circ)+\cos155^\circ$
$=\ 2\times\frac{1}{2}\cos25^\circ+\cos155^\circ$
$=\ \cos25^\circ+\cos155^\circ$
$=\ 2\cos\Big(\frac{25^\circ+155^\circ}{2}\Big)\cos\Big(\frac{25^\circ-155^\circ}{2}\Big)$
$=\ 2\cos90^\circ\cos65^\circ$
$=\ 0$

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