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)

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