If $\sin \theta + \sin^2 \theta = 1$,then $\cos^2 \theta + \cos^4 \theta = \dots$

Write 'True' or 'False' and justify your answer.
The value of the expression $(\cos^{2} 23^{\circ} - \sin^{2} 67^{\circ})$ is positive.

Prove the following identities,where the angles involved are acute angles for which the expressions are defined:
$\left(\frac{1+\tan ^{2} A}{1+\cot ^{2} A}\right)=\left(\frac{1-\tan A}{1-\cot A}\right)^{2}=\tan ^{2} A$

$\tan (65^\circ - \theta) - \cot (25^\circ + \theta) - \sec (55^\circ - \theta) + \operatorname{cosec}(35^\circ + \theta) = \ldots \ldots \ldots \ldots$ (where,$0 < \theta < 25^\circ$)

If $\sin 3A = \cos(A - 26^{\circ})$,where $3A$ is an acute angle,find the value of $A$.

If $3 \theta$ is the measure of an acute angle and $\sin 3 \theta = \cos (\theta - 26^{\circ})$,then the value of $\theta$ is $\ldots \ldots \ldots \ldots$ (in $^{\circ}$)