We need to simplify the expression:

$(\sin 60^\circ)(\cos 30^\circ) + (\cos 60^\circ)(\sin 30^\circ)$

Step-by-step Solution:

1. Substitute the trigonometric values for the specific angles:

   • $\sin 60^\circ = \frac{\sqrt{3}}{2}$
   • $\cos 30^\circ = \frac{\sqrt{3}}{2}$
   • $\cos 60^\circ = \frac{1}{2}$
   • $\sin 30^\circ = \frac{1}{2}$

   So, the expression becomes:

   $\left( \frac{\sqrt{3}}{2} \right) \left( \frac{\sqrt{3}}{2} \right) + \left( \frac{1}{2} \right) \left( \frac{1}{2} \right)$

2. Simplify each term:

   • $\frac{\sqrt{3}}{2} \times \frac{\sqrt{3}}{2} = \frac{3}{4}$
   • $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$

   Now the expression becomes:

   $\frac{3}{4} + \frac{1}{4}$

3. Add the fractions:

   $\frac{3}{4} + \frac{1}{4} = \frac{4}{4} = 1$

Final Answer:

$(\sin 60^\circ)(\cos 30^\circ) + (\cos 60^\circ)(\sin 30^\circ) = 1$

This expression is equal to 1.

Would you like a deeper explanation of any of these steps?

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Here are 5 related questions to practice:

1. Simplify $(\sin 45^\circ)(\cos 45^\circ) + (\cos 45^\circ)(\sin 45^\circ)$.
2. What is the value of $\sin(30^\circ + 60^\circ)$?
3. Prove the identity: $\sin(A + B) = \sin A \cos B + \cos A \sin B$.
4. Evaluate $\sin 30^\circ + \cos 60^\circ$.
5. Simplify the expression $(\sin 90^\circ)(\cos 0^\circ)$.

Tip: Trigonometric identities like $\sin(A + B)$ can help simplify complex expressions!