Question

Find the value of $\cos 15^{\circ }-\sin 15^{\circ }$ ?

• A. $2\sqrt{2}$
• B. $\frac{\sqrt{3}}{2}$
• C. $\frac{2\sqrt{3}}{2}$
• D. $\frac{1}{\sqrt{2}}$

Answer 1

Answer: (D)

$\cos 15^{\circ }=\cos (45^{\circ }-30^{\circ })$
$=\cos 45^{\circ }\cos 30^{\circ }+\sin 45^{\circ }\sin 30^{\circ }$
$=\frac{1}{\sqrt{2}}\times \frac{\sqrt{3}}{2}+\frac{1}{\sqrt{2}}\times \frac{1}{2}=\frac{\sqrt{3}+1}{2\sqrt{2}}$
$\sin 15^{\circ }=\sin (45^{\circ }-30^{\circ })$
$=\sin 45^{\circ }\cos 30^{\circ }-\cos 45^{\circ }\sin 30^{\circ }$
$=\frac{1}{\sqrt{2}}\times \frac{\sqrt{3}}{2}-\frac{1}{\sqrt{2}}\times \frac{1}{2}=\frac{\sqrt{3}-1}{2\sqrt{2}}$
$\text{ So, }\cos 15^{\circ }-\sin 15^{\circ }$
$=\frac{\sqrt{3}+1}{2\sqrt{2}}=\frac{\sqrt{3}+1}{2\sqrt{2}}=\frac{2}{2\sqrt{2}}=\frac{1}{\sqrt{2}}$