What is (sin 9° − cos 9°) equal to?

Correct option is A

Solution:

(sin 9° − cos 9°)

sin A − cos A = √2 × sin(A − 45°)
​$$\sin 9^\circ - \cos 9^\circ = \sqrt{2} \cdot \sin(9^\circ - 45^\circ) \\= \sqrt{2} \cdot \sin(-36^\circ) = -\sqrt{2} \cdot \sin(36^\circ) \\= -\sqrt{2} \cdot \frac{\sqrt{10 - 2\sqrt{5}}}{4} \\\ \\= -\frac{\sqrt{2(10 - 2\sqrt{5})}}{4} \\\ \\= -\frac{\sqrt{20 - 4\sqrt{5}}}{4} \\\ \\= -\frac{1}{4} \cdot \sqrt{4(5 - \sqrt{5})} \\\ \\= -\frac{1}{4} \cdot 2\sqrt{5 - \sqrt{5}} \\\ \\= \boxed{-\frac{\sqrt{5 - \sqrt{5}}}{2}}$$​​