Question

The value of $2\sin \left(12^{\circ }\right)-\sin \left(72^{\circ }\right)$ is :

• A. $\frac{\sqrt{5}(1-\sqrt{3})}{4}$
• B. $\frac{1-\sqrt{5}}{8}$
• C. $\frac{\sqrt{3}(1-\sqrt{5})}{2}$
• D. $\frac{\sqrt{3}(1-\sqrt{5})}{4}$

Answer 1

Answer: (D)

$\sin 12^{\circ }+\sin 12^{\circ }-\sin 72^{\circ }$
$=\sin 12^{\circ }-2\cos 42^{\circ }\sin 30^{\circ }$
$=\sin 12^{\circ }-\sin 48^{\circ }$
$=-2\cos 30^{\circ }\sin 18^{\circ }$
$=-2\times \frac{\sqrt{3}}{2}\times \frac{\sqrt{5}-1}{4}$
$=\frac{\sqrt{3}}{4}(1-\sqrt{5})$