$\frac{1+\tan ^{2} A}{1+\cot ^{2} A} = \dots$

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.
$\sin \theta = \cos \theta$ for all values of $\theta$.

Prove that $\frac{\cot A - \cos A}{\cot A + \cos A} = \frac{\operatorname{cosec} A - 1}{\operatorname{cosec} A + 1}$.

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

Evaluate the following:
$2 \tan ^{2} 45^{\circ}+\cos ^{2} 30^{\circ}-\sin ^{2} 60^{\circ}$