The value of 52^ + 68^ + 172^ is - Vidyadip

MCQ

The value of $\cos52^\circ+\cos68^\circ+\cos172^\circ\text{ is }$

✓
$0$
B
$1$
C
$2$
D
$\frac{3}{2}$

✓

Answer

Correct option: A.

$0$

$\cos52^\circ+\cos68^\circ+\cos172^\circ$
$=\ 2\cos\Big(\frac{52^\circ+68^\circ}{2}\Big)\cos\Big(\frac{52^\circ-68^\circ}{2}\Big)+\cos172^\circ$ $\Big[\because\ \cos\text{A}+\cos\text{B}=2\cos\Big(\frac{\text{A+B}}{2}\Big)\cos\Big(\frac{\text{A}-\text{B}}{2}\Big)\Big]$
$=\ 2\cos60^\circ\cos(-8^\circ)+\cos172^\circ$
$=\ 2\times\frac{1}{2}\cos8^\circ+\cos172^\circ$
$=\ \cos8^\circ+\cos172^\circ$
$=\ 2\cos\Big(\frac{8^\circ+172^\circ}{2}\Big)\cos\Big(\frac{8^\circ-172^\circ}{2}\Big)$
$=\ 2\cos90^\circ\cos82^\circ$
$=\ 0$

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