Evaluate the following:
$\frac{\sin 30^{\circ}+\tan 45^{\circ}-\operatorname{cosec} 60^{\circ}}{\sec 30^{\circ}+\cos 60^{\circ}+\cot 45^{\circ}}$

Substitute the trigonometric values:
$\sin 30^{\circ} = \frac{1}{2}$,$\tan 45^{\circ} = 1$,$\operatorname{cosec} 60^{\circ} = \frac{2}{\sqrt{3}}$,$\sec 30^{\circ} = \frac{2}{\sqrt{3}}$,$\cos 60^{\circ} = \frac{1}{2}$,$\cot 45^{\circ} = 1$.
$\frac{\frac{1}{2}+1-\frac{2}{\sqrt{3}}}{\frac{2}{\sqrt{3}}+\frac{1}{2}+1} = \frac{\frac{3}{2}-\frac{2}{\sqrt{3}}}{\frac{3}{2}+\frac{2}{\sqrt{3}}}$
$= \frac{\frac{3\sqrt{3}-4}{2\sqrt{3}}}{\frac{3\sqrt{3}+4}{2\sqrt{3}}} = \frac{3\sqrt{3}-4}{3\sqrt{3}+4}$
Rationalizing the denominator:
$= \frac{(3\sqrt{3}-4)(3\sqrt{3}-4)}{(3\sqrt{3}+4)(3\sqrt{3}-4)} = \frac{(3\sqrt{3})^2 + 4^2 - 2(3\sqrt{3})(4)}{(3\sqrt{3})^2 - 4^2}$
$= \frac{27 + 16 - 24\sqrt{3}}{27 - 16} = \frac{43 - 24\sqrt{3}}{11}$