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

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

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

Show that:

$(i)$ $\tan 48^{\circ} \tan 23^{\circ} \tan 42^{\circ} \tan 67^{\circ}=1$

$(ii)$ $\cos 38^{\circ} \cos 52^{\circ}-\sin 38^{\circ} \sin 52^{\circ}=0$

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

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

$(1+\tan \theta+\sec \theta)(1+\cot \theta-\operatorname{cosec} \theta)=……….$

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

$(\operatorname{cosec} A-\sin A)(\sec A-\cos A)=\frac{1}{\tan A+\cot A}$