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

Write 'True' or 'False' and justify your answer.
The value of the expression $(\sin 80^{\circ} - \cos 80^{\circ})$ is negative.

If $\cos \theta = \frac{15}{17}$,then the value of $\operatorname{cosec} \theta + \cot \theta$ is .........

Show that $\frac{\cos ^{2}\left(45^{\circ}+\theta\right)+\cos ^{2}\left(45^{\circ}-\theta\right)}{\tan \left(60^{\circ}+\theta\right) \tan \left(30^{\circ}-\theta\right)}=1$

If $\operatorname{cosec} \theta = \frac{2}{\sqrt{3}}$,then $\theta = \ldots$ (in $^\circ$)

If $\operatorname{cosec} \theta + \cot \theta = p$,then prove that $\cos \theta = \frac{p^{2} - 1}{p^{2} + 1}$.