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 ：$$\sin {45}^{\circ }\cos {30}^{\circ }+\cos {60}^{\circ }\sin {60}^{\circ }=\boxed{{\displaystyle 1}}$$