I calculated $\mathrm{sin}75}^{\circ$ as $\frac{1}{2\sqrt{2}}+\frac{\sqrt{3}}{2\sqrt{2}}$, but the answer is $\frac{\sqrt{2}+\sqrt{6}}{4}$. What went wrong?

I calculated the exact value of $\mathrm{sin}75}^{\circ$ as follows:

${\mathrm{sin}75}^{\circ}=\mathrm{sin}({30}^{\circ}+{45}^{\circ})$

$=\mathrm{sin}30\xb0\mathrm{cos}45\xb0+\mathrm{cos}30\xb0\mathrm{sin}45\xb0$

$=\frac{1}{2}\xb7\frac{1}{\sqrt{2}}+\frac{\sqrt{3}}{2}\xb7\frac{1}{\sqrt{2}}$

$=\frac{1}{2\sqrt{2}}+\frac{\sqrt{3}}{2\sqrt{2}}$

The actual answer is

$\frac{\sqrt{2}+\sqrt{6}}{4}$