If $\sin 3A = \cos(A - 26^{\circ})$,where $3A$ is an acute angle,find the value of $A$.

Evaluate the following:
$\frac{5 \cos ^{2} 60^{\circ}+4 \sec ^{2} 30^{\circ}-\tan ^{2} 45^{\circ}}{\sin ^{2} 30^{\circ}+\cos ^{2} 30^{\circ}}$ (in $/12$)

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

If $\cot \theta = \frac{7}{8},$ evaluate:
$(i) \frac{(1+\sin \theta)(1-\sin \theta)}{(1+\cos \theta)(1-\cos \theta)}$
$(ii) \cot^2 \theta$

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

Consider $\triangle ACB$,right-angled at $C$,in which $AB = 29$ units,$BC = 21$ units and $\angle ABC = \theta$. Determine the values of:
$(i)$ $\cos^2 \theta + \sin^2 \theta$
$(ii)$ $\cos^2 \theta - \sin^2 \theta$