$\frac{1-\tan ^{2} 45^{\circ}}{1+\tan ^{2} 45^{\circ}}=$

Show that:
$(i)$ $\tan 48^{\circ} \tan 23^{\circ} \tan 42^{\circ} \tan 67^{\circ} = 1$
$(ii)$ $\cos 38^{\circ} \cos 52^{\circ} - \sin 38^{\circ} \sin 52^{\circ} = 0$

State whether the following is true or false. Justify your answer.
$\cot A$ is not defined for $A = 0^{\circ}$.

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

Prove the following identity,where the angles involved are acute angles for which the expressions are defined:
$\frac{\cos A-\sin A+1}{\cos A+\sin A-1}=\operatorname{cosec} A+\cot A$,using the identity $\operatorname{cosec}^{2} A=1+\cot ^{2} A$.

Express the ratios $\cos A$,$\tan A$,and $\sec A$ in terms of $\sin A$.