Find the exact value of the expression.
$\sin15\degree$

• A. $\frac{\sqrt2-\sqrt6}{4}$
• B. $\frac{\sqrt6-\sqrt2}{4}$
• C. $\frac{\sqrt2-\sqrt6}{2}$
• D. $\frac{\sqrt4}{4}$

Answer: (B)

Verified step by step guidance

1. Recognize that the expression involves the sine of an angle, specifically \( \sin 15^\circ \). This angle can be expressed using the angle subtraction identity: \( 15^\circ = 45^\circ - 30^\circ \).
2. Apply the sine subtraction formula: \( \sin(a - b) = \sin a \cos b - \cos a \sin b \). For \( a = 45^\circ \) and \( b = 30^\circ \), substitute these values into the formula.
3. Calculate \( \sin 45^\circ \), \( \cos 45^\circ \), \( \sin 30^\circ \), and \( \cos 30^\circ \) using known values: \( \sin 45^\circ = \cos 45^\circ = \frac{\sqrt{2}}{2} \), \( \sin 30^\circ = \frac{1}{2} \), and \( \cos 30^\circ = \frac{\sqrt{3}}{2} \).
4. Substitute these trigonometric values into the sine subtraction formula: \( \sin 15^\circ = \left( \frac{\sqrt{2}}{2} \right) \left( \frac{\sqrt{3}}{2} \right) - \left( \frac{\sqrt{2}}{2} \right) \left( \frac{1}{2} \right) \).
5. Simplify the expression: \( \sin 15^\circ = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = \frac{\sqrt{6} - \sqrt{2}}{4} \). This matches one of the given answer choices.