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

$\sin 2A = 2 \sin A$ is true when $A =$ (in $^{\circ}$)

State whether the following is true or false. Justify your answer.
$\cot A$ is not defined for $A = 0^{\circ}$.

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

Prove that $\frac{\sin \theta-\cos \theta+1}{\sin \theta+\cos \theta-1}=\frac{1}{\sec \theta-\tan \theta},$ using the identity $\sec ^{2} \theta=1+\tan ^{2} \theta.$

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