Question

The value of \(cos15^{\circ}-sin15^{\circ}\) is ......

• A. \(\sqrt{2}\)
• B. \(-\sqrt{2}\)
• C. \(\frac{1}{\sqrt{2}}\)
• D. \(-\frac{1}{\sqrt{2}}\)

Hint:

Write cos x minus sin x as root 2 cos(x + 45 degrees).

Answer 1

The Correct Option is C

Step 1: Use exact values:
$\cos 15^{\circ} = \frac{\sqrt6+\sqrt2}{4}$ and $\sin 15^{\circ} = \frac{\sqrt6-\sqrt2}{4}$.

Step 2: Subtract:
\[ \frac{\sqrt6+\sqrt2}{4} - \frac{\sqrt6-\sqrt2}{4} = \frac{2\sqrt2}{4} = \frac{\sqrt2}{2} = \frac{1}{\sqrt2} \]

Step 3: Sign check:
The result is positive, which confirms options (B) and (D) cannot be correct.

Final Answer:
The value is 1 over root 2, option (C). \[ \boxed{\frac{1}{\sqrt{2}}} \]