test

$$ \begin{split} \sin(30+45) &= \sin(30)\cos(45) + \sin(45)\cos(30)\ &=\frac{1}{2} * \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} * \frac{\sqrt{3}}{2}\ &=\frac{\sqrt{2}+\sqrt{6}}{4} \end{split} $$

$$ \begin{split} \cos(30+45) &= \cos(30)\cos(45) + \sin(45)\sin(30)\ &=\frac{\sqrt{3}}{2} * \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} * \frac{1}{2}\ &=\frac{\sqrt{6}+\sqrt{2}}{4} \end{split} $$

$$ \begin{split} \tan(30+45) &= \frac{\tan(30) + \tan(45)}{1 - \tan(30)\tan(45)}\ &=\frac{\frac{1}{\sqrt{3}} + 1}{1 - \frac{1}{\sqrt{3}} \cdot 1}\ &={(\frac{1 +\sqrt{3}}{\sqrt{3}})} \div {(\frac{\sqrt{3} - 1}{\sqrt{3}})}\ &={\frac{1 +\sqrt{3}}{\sqrt{3} - 1}}\ &={\frac{4 + 2\sqrt{3}}{2}}\ &={2 + \sqrt{3}}\ \end{split} $$