$\text{a) }\sin {15}^{\circ }=\sin \left({45}^{\circ }-{30}^{\circ }\right)=\sin {45}^{\circ }\cos {30}^{\circ }-\cos {45}^{\circ }\sin {30}^{\circ }=\frac{\sqrt{2}}{2}\cdot \frac{\sqrt{3}}{2}-\frac{\sqrt{2}}{2}\cdot \frac{1}{2}=\frac{\sqrt{6}-\sqrt{2}}{4}$  

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$\text{b) }\cos {15}^{\circ }=\cos \left({45}^{\circ }-{30}^{\circ }\right)=\cos {45}^{\circ }\cos {30}^{\circ }+\sin {45}^{\circ }\sin {30}^{\circ }=\frac{\sqrt{2}}{2}\cdot \frac{\sqrt{3}}{2}+\frac{\sqrt{2}}{2}\cdot \frac{1}{2}=\frac{\sqrt{6}+\sqrt{2}}{4}$  

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$\text{c) }\mathrm{tg}{75}^{\circ }=\frac{\sin {75}^{\circ }}{\cos {75}^{\circ }}=\frac{\sin \left({90}^{\circ }-{15}^{\circ }\right)}{\cos \left({90}^{\circ }-{15}^{\circ }\right)}=\frac{\cos {15}^{\circ }}{\sin {15}^{\circ }}=\frac{\sqrt{6}+\sqrt{2}}{4}\cdot \frac{4}{\sqrt{6}-\sqrt{2}}=\frac{\sqrt{6}+\sqrt{2}}{\sqrt{6}-\sqrt{2}}=\frac{{\left(\sqrt{6}+\sqrt{2}\right)}^2}{\left(\sqrt{6}-\sqrt{2}\right)\left(\sqrt{6}+\sqrt{2}\right)}=\frac{6+2\sqrt{12}+2}{6-2}=\frac{8+4\sqrt{3}}{4}=2+\sqrt{3}$  

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$\text{d) }\sin {105}^{\circ }=\sin \left({90}^{\circ }-\left(-{15}^{\circ }\right)\right)=\cos \left(-{15}^{\circ }\right)=\cos {15}^{\circ }\stackrel{\text{b)}}{=}\frac{\sqrt{6}+\sqrt{2}}{4}$  

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$\text{e) }\cos {105}^{\circ }=\cos \left({90}^{\circ }-\left(-{15}^{\circ }\right)\right)=\sin \left(-{15}^{\circ }\right)=-\sin {15}^{\circ }\stackrel{\text{a)}}{=}-\frac{\sqrt{6}-\sqrt{2}}{4}=\frac{\sqrt{2}-\sqrt{6}}{4}$