$\alpha$	${30}^{\circ }$	${45}^{\circ }$	${60}^{\circ }$
$\sin \alpha$	$\frac{1}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{3}}{2}$
$\cos \alpha$	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{1}{2}$
$\text{tg}\alpha$	$\frac{\sqrt{3}}{3}$	$1$	$\sqrt{3}$

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$\text{a)}\sqrt{\underset{1}{\underbrace{{\sin }^2{60}^{\circ }+{\cos }^2{60}^{\circ }}}+{\text{tg}}^2{60}^{\circ }}=\sqrt{1+{\left(\sqrt{3}\right)}^2}=\sqrt{1+3}=\sqrt{4}=2$  

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$\text{b)}\left(\sin {45}^{\circ }+\cos {45}^{\circ }\right):\left(\sin {30}^{\circ }+\cos {60}^{\circ }\right)=\left(\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}\right):\left(\frac{1}{2}+\frac{1}{2}\right)=\sqrt{2}:1=\sqrt{2}$  

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$\text{c)}\frac{\text{tg}{30}^{\circ }\cos {30}^{\circ }}{\sin {30}^{\circ }\text{tg}{45}^{\circ }}=\frac{\frac{\sqrt{3}}{3}\cdot \frac{\sqrt{3}}{2}}{\frac{1}{2}\cdot 1}=\frac{\frac{1}{2}}{\frac{1}{2}}=1$