$\operatorname{cosec} 40^{\circ} = \ldots \ldots \ldots \ldots$

State whether the following are true or false. Justify your answer.
$(i)$ The value of $\tan A$ is always less than $1$.
$(ii)$ $\sec A = \frac{12}{5}$ for some value of angle $A$.

Which of the following groups truly matches the data of Part $I$ with the data of Part $II$?

Part $I$	Part $II$
$1.$ $\cos(90^\circ - \theta)$	$a.$ $\sec \theta$
$2.$ $\cot(90^\circ - \theta)$	$b.$ $\sin \theta$
$3.$ $\operatorname{cosec}(90^\circ - \theta)$	$c.$ $1$
	$d.$ $\tan \theta$

In a right triangle $ABC$,right-angled at $B$. If $\tan A = 1$,then verify that $2 \sin A \cos A = 1$.

Given $\sec \theta = \frac{13}{12}$,calculate all other trigonometric ratios.

$2A$ is the measure of an acute angle and $\sec 2A = \operatorname{cosec}(A - 42^\circ)$,then the value of $A$ is $\ldots \ldots \ldots \ldots$ (in $^\circ$)