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)

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