In $\Delta ABC$,$m \angle C = 90^{\circ}$ and $\cos B = \frac{1}{2}$,then $\operatorname{cosec} A = \ldots$

Write 'True' or 'False' and justify your answer.
$\tan \theta$ increases faster than $\sin \theta$ as $\theta$ increases for $0^\circ < \theta < 90^\circ$.

Write 'True' or 'False' and justify your answer.
The value of the expression $(\sin 80^{\circ} - \cos 80^{\circ})$ is negative.

If $\cot \theta = \frac{a}{b}$,then $\frac{\cos \theta - \sin \theta}{\cos \theta + \sin \theta} = \ldots$

State whether the following are true or false. Justify your answer.
$(i)$ $\cos A$ is the abbreviation used for the cosecant of angle $A$.
$(ii)$ $\cot A$ is the product of $\cot$ and $A$.
$(iii)$ $\sin \theta = \frac{4}{3}$ for some angle $\theta$.

Write 'True' or 'False' and justify your answer.
The value of $2 \sin \theta$ can be $(a + \frac{1}{a}),$ where $a$ is a positive number,and $a \neq 1$.