In triangle $ABC ,$ right -angled at $B ,$ if $\tan A =\frac{1}{\sqrt{3}},$ find the value of:

$(i)$ $\sin A \cos C+\cos A \sin C$

$(ii)$ $\cos A \cos C-\sin A \sin C$

Prove the following identities, where the angles involved are acute angles for which the expressions are defined.

$(\sin A+\operatorname{cosec} A)^{2}+(\cos A+\sec A)^{2}=7+\tan ^{2} A+\cot ^{2} A$

If $\cot \theta=\frac{7}{8},$ evaluate:

$(i)$ $\frac{(1+\sin \theta)(1-\sin \theta)}{(1+\cos \theta)(1-\cos \theta)}$

$(ii)$ $\cot ^{2} \theta$

Prove the following identities, where the angles involved are acute angles for which the expressions are defined.

$\left(\frac{1+\tan ^{2} A}{1+\cot ^{2} A}\right)=\left(\frac{1-\tan A}{1-\cot A}\right)^{2}=\tan ^{2} A$

State whether the following are true or false. Justify your answer.

The value of $\sin \theta$ increases as $\theta$ increases.