If $\sin ^{2}(3 x+30^{\circ})+\cos ^{2}(2 x+45^{\circ})=1$,then $x = \dots$ (in $^{\circ}$)

$\sec (90^\circ - \theta) = \dots$

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 $^{\circ}$)

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

Prove that $\sin^{6} \theta + \cos^{6} \theta + 3 \sin^{2} \theta \cos^{2} \theta = 1$.

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