If $\tan 2A = \cot(A - 18^{\circ})$,where $2A$ is an acute angle,find the value of $A$ (in $^{\circ}$).

In $\triangle ABC$,right-angled at $B$,$AB = 5 \, cm$ and $\angle ACB = 30^{\circ}$. Determine the lengths of the sides $BC$ and $AC$.

In $\triangle PQR$, right-angled at $Q$, $PR + QR = 25 \, cm$ and $PQ = 5 \, cm$. Determine the values of $\sin P, \cos P$ and $\tan P$.

$\sin 2A = 2 \sin A$ is true when $A =$ (in $^{\circ}$)

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 $\theta$.

State whether the following is true or false. Justify your answer.
The value of $\cos \theta$ increases as $\theta$ increases.