Simplifying Trigonometric Expression: \(\sin(60^\circ+x) + \sin(60^\circ-x)\)

This problem requires simplifying the sum of two sine functions using trigonometric identities.

Applying Sum-to-Product Identity (or Sum/Difference Formulas)

We can use the angle addition and subtraction formulas for sine:

• \(\sin(A+B) = \sin A \cos B + \cos A \sin B\)
• \(\sin(A-B) = \sin A \cos B - \cos A \sin B\)

Let \(A = 60^\circ\) and \(B = x\). Applying these formulas to the expression:

1. Expand \(\sin(60^\circ+x)\):

   \(\sin(60^\circ+x) = \sin 60^\circ \cos x + \cos 60^\circ \sin x\)

2. Expand \(\sin(60^\circ-x)\):

   \(\sin(60^\circ-x) = \sin 60^\circ \cos x - \cos 60^\circ \sin x\)

3. Add the two expanded expressions:

   \((\sin 60^\circ \cos x + \cos 60^\circ \sin x) + (\sin 60^\circ \cos x - \cos 60^\circ \sin x)\)

4. Combine like terms. Notice that \(\cos 60^\circ \sin x\) and \(-\cos 60^\circ \sin x\) cancel out:

   \(= \sin 60^\circ \cos x + \sin 60^\circ \cos x\)

   \(= 2 \sin 60^\circ \cos x\)

5. Substitute the known value of \(\sin 60^\circ = \frac{\sqrt{3}}{2}\):

   \(= 2 \left( \frac{\sqrt{3}}{2} \right) \cos x\)

6. Simplify the expression:

   \(= \sqrt{3} \cos x\)

Final Result

The value of \(\sin(60^\circ+x) + \sin(60^\circ-x)\) is \(\sqrt{3} \cos x\). This corresponds to Option 3.