If $\sec 4A = \operatorname{cosec}(A - 20^\circ)$,where $4A$ is an acute angle,then the value of $A$ is $\ldots \ldots \ldots \ldots .$ (in $^\circ$)

$(\sin \theta+\cos \theta)^{2}+(\sin \theta-\cos \theta)^{2} = \dots$

$\sec 55^{\circ} \cdot \sin 35^{\circ} + \cos 35^{\circ} \cdot \operatorname{cosec} 55^{\circ} = \ldots \ldots \ldots \ldots$

Express the ratios $\cos A$,$\tan A$,and $\sec A$ in terms of $\sin A$.

Prove the following identity,where the angles involved are acute angles for which the expressions are defined:
$(\sin A + \operatorname{cosec} A)^2 + (\cos A + \sec A)^2 = 7 + \tan^2 A + \cot^2 A$

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