Using sum and difference identities, what is the exact value of $\sin (75°)$ ?

• A. $\frac{\sqrt{6}-\sqrt{2}}{4}$
• B. $\frac{\sqrt{2}}{2}$
• C. $\frac{\sqrt{3}}{2}$
• D. $\frac{\sqrt{6}+\sqrt{2}}{4}$

Answer: (D)

Verified step by step guidance

1. Recognize that 75° can be expressed as the sum of two special angles whose sine and cosine values are well known, for example, 75° = 45° + 30°.
2. Recall the sine sum identity: \(\sin(A + B) = \sin A \cos B + \cos A \sin B\).
3. Substitute \(A = 45^\circ\) and \(B = 30^\circ\) into the identity to get \(\sin 75^\circ = \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ\).
4. Use the exact values of sine and cosine for 30° and 45°: \(\sin 45^\circ = \frac{\sqrt{2}}{2}\), \(\cos 45^\circ = \frac{\sqrt{2}}{2}\), \(\sin 30^\circ = \frac{1}{2}\), and \(\cos 30^\circ = \frac{\sqrt{3}}{2}\).
5. Substitute these values into the expression and simplify the resulting terms by combining like terms and factoring to reach the exact value of \(\sin 75^\circ\).