Дано:
1) $\cos 60^\circ - \mathrm{tg}^2 45^\circ + \frac{3}{4} \mathrm{tg}^2 30^\circ + 4 \cos^2 30^\circ - \sin 30^\circ$;
2) $\cos^2 30^\circ + 2 \sin 30^\circ - \mathrm{ctg}^2 45^\circ + \mathrm{ctg}^2 30^\circ + \cos 60^\circ$;
3) $\mathrm{ctg}^2 45^\circ + \cos 60^\circ - \sin^2 60^\circ + \frac{3}{4} \mathrm{ctg}^2 60^\circ$;
4) $\mathrm{tg}^2 30^\circ - \mathrm{tg} 45^\circ - \cos^2 30^\circ + 2 \sin 60^\circ$.

Решение:
1) $\cos 60^\circ - \mathrm{tg}^2 45^\circ + \frac{3}{4} \mathrm{tg}^2 30^\circ + 4 \cos^2 30^\circ - \sin 30^\circ = \frac{1}{2} - 1^2 + \frac{3}{4} \cdot \left(\frac{1}{\sqrt{3}}\right)^2 + 4 \cdot \left(\frac{\sqrt{3}}{2}\right)^2 - \frac{1}{2} = \frac{1}{2} - 1 + \frac{3}{4} \cdot \frac{1}{3} + 4 \cdot \frac{3}{4} - \frac{1}{2} = \frac{1}{2} - 1 + \frac{1}{4} + 3 - \frac{1}{2} = -1 + \frac{1}{4} + 3 = 2 + \frac{1}{4} = \frac{8}{4} + \frac{1}{4} = \frac{9}{4} = 2.25$.

2) $\cos^2 30^\circ + 2 \sin 30^\circ - \mathrm{ctg}^2 45^\circ + \mathrm{ctg}^2 30^\circ + \cos 60^\circ = \left(\frac{\sqrt{3}}{2}\right)^2 + 2 \cdot \frac{1}{2} - 1^2 + (\sqrt{3})^2 + \frac{1}{2} = \frac{3}{4} + 1 - 1 + 3 + \frac{1}{2} = \frac{3}{4} + 3 + \frac{1}{2} = \frac{3}{4} + \frac{12}{4} + \frac{2}{4} = \frac{17}{4} = 4.25$.

3) $\mathrm{ctg}^2 45^\circ + \cos 60^\circ - \sin^2 60^\circ + \frac{3}{4} \mathrm{ctg}^2 60^\circ = 1^2 + \frac{1}{2} - \left(\frac{\sqrt{3}}{2}\right)^2 + \frac{3}{4} \cdot \left(\frac{1}{\sqrt{3}}\right)^2 = 1 + \frac{1}{2} - \frac{3}{4} + \frac{3}{4} \cdot \frac{1}{3} = 1 + \frac{1}{2} - \frac{3}{4} + \frac{1}{4} = 1 + \frac{1}{2} - \frac{2}{4} = 1 + \frac{1}{2} - \frac{1}{2} = 1$.

4) $\mathrm{tg}^2 30^\circ - \mathrm{tg} 45^\circ - \cos^2 30^\circ + 2 \sin 60^\circ = \left(\frac{1}{\sqrt{3}}\right)^2 - 1 - \left(\frac{\sqrt{3}}{2}\right)^2 + 2 \cdot \frac{\sqrt{3}}{2} = \frac{1}{3} - 1 - \frac{3}{4} + \sqrt{3} = \frac{1}{3} - \frac{4}{4} - \frac{3}{4} + \sqrt{3} = \frac{1}{3} - \frac{7}{4} + \sqrt{3} = \frac{4}{12} - \frac{21}{12} + \sqrt{3} = -\frac{17}{12} + \sqrt{3}$.

Ответ:
1) $\frac{9}{4} = 2.25$;
2) $\frac{17}{4} = 4.25$;
3) $1$;
4) $-\frac{17}{12} + \sqrt{3}$.