Use table for trigonometric function values of some common angles and simplify the resulting expression. \[ \sin 45^{\circ} \cos 30^{\circ}+\cos 60^{\circ} \sin 60^{\circ} \]

Step 1 ：From the angle addition formula, we have

Step 2 ：\[\sin (A+B) = \sin A \cos B + \cos A \sin B\]

Step 3 ：Substituting \(A = 45^\circ\) and \(B = 30^\circ\) into the formula, we get

Step 4 ：\[\sin 75^\circ = \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ\]

Step 5 ：Substituting \(A = 60^\circ\) and \(B = 60^\circ\) into the formula, we get

Step 6 ：\[\sin 120^\circ = \sin 60^\circ \cos 60^\circ + \cos 60^\circ \sin 60^\circ\]

Step 7 ：Since \(\sin 75^\circ = \sin 120^\circ\), we can equate the two expressions to get

Step 8 ：\[\sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ = \sin 60^\circ \cos 60^\circ + \cos 60^\circ \sin 60^\circ\]

Step 9 ：Subtracting \(\sin 60^\circ \cos 60^\circ + \cos 60^\circ \sin 60^\circ\) from both sides, we get

Step 10 ：\[\sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ - \sin 60^\circ \cos 60^\circ - \cos 60^\circ \sin 60^\circ = 0\]

Step 11 ：Substituting the values of \(\sin 45^\circ = \frac{1}{\sqrt{2}}\), \(\cos 30^\circ = \frac{\sqrt{3}}{2}\), \(\cos 45^\circ = \frac{1}{\sqrt{2}}\), \(\sin 30^\circ = \frac{1}{2}\), \(\sin 60^\circ = \frac{\sqrt{3}}{2}\), and \(\cos 60^\circ = \frac{1}{2}\) into the equation, we get

Step 12 ：\[\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2} + \frac{1}{\sqrt{2}} \cdot \frac{1}{2} - \frac{\sqrt{3}}{2} \cdot \frac{1}{2} - \frac{1}{2} \cdot \frac{\sqrt{3}}{2} = 0\]

Step 13 ：Simplifying the equation, we get

Step 14 ：\[\frac{\sqrt{3}}{4\sqrt{2}} + \frac{1}{4\sqrt{2}} - \frac{1}{4} - \frac{\sqrt{3}}{4} = 0\]

Step 15 ：Combining like terms, we get

Step 16 ：\[\frac{\sqrt{3} + 1}{4\sqrt{2}} - \frac{1 + \sqrt{3}}{4} = 0\]

Step 17 ：Multiplying both sides by \(4\sqrt{2}\), we get

Step 18 ：\[\sqrt{3} + 1 - \sqrt{2}(1 + \sqrt{3}) = 0\]

Step 19 ：Solving for \(\sqrt{2}\), we get

Step 20 ：\[\sqrt{2} = \frac{\sqrt{3} + 1}{1 + \sqrt{3}}\]

Step 21 ：Substituting \(\sqrt{2} = \frac{\sqrt{3} + 1}{1 + \sqrt{3}}\) into the original equation, we get

Step 22 ：\[\sin 45^\circ \cos 30^\circ + \cos 60^\circ \sin 60^\circ = \frac{\sqrt{3} + 1}{1 + \sqrt{3}}\]

Step 23 ：Simplifying the equation, we get

Step 24 ：\[\sin 45^\circ \cos 30^\circ + \cos 60^\circ \sin 60^\circ = \boxed{1}\]