Question

The value of $\sin 55^{\circ}-\sin 19^{\circ}+\sin 53^{\circ}-\sin 17^{\circ}$ is always equal to

• A. $\cos 1^{\circ}$
• B. $\sin 1^{\circ}$
• C. $\tan 1^{\circ}$ $
• D. -\cos 1^{\circ}$

Answer 1

Answer: (A)

$\sin 55^{\circ}-\sin 19^{\circ}+\sin 53^{\circ}-\sin 17^{\circ}$
$=2 \cos 37^{\circ} \sin 18^{\circ}+2 \cos 35^{\circ} \sin 18^{\circ}$
$=2 \sin 18^{\circ}\left(\cos 37^{\circ}+\cos 35^{\circ}\right)$
$=2 \frac{\sqrt{5}-1}{4}\left(2 \cos 36^{\circ} \cos 1^{\circ}\right)=(\sqrt{5}-1) \frac{(\sqrt{5}+1)}{4} \cos 1^{\circ}=\cos 1^{\circ}$