If $\theta$ is in the third quadrant,then $\sqrt{4 \sin ^4 \theta+\sin ^2 2 \theta}+4 \cos ^2\left(\frac{\pi}{4}-\frac{\theta}{2}\right)=$

If $\cos x + \cos^2 x = 1$,then the value of $\sin^2 x + \sin^4 x$ is

The value of the expression $\frac{2(\sin 1^{\circ} + \sin 2^{\circ} + \sin 3^{\circ} + \dots + \sin 89^{\circ})}{2(\cos 1^{\circ} + \cos 2^{\circ} + \dots + \cos 44^{\circ}) + 1}$ equals

If $\frac{\cos x}{a} = \frac{\cos (x + \theta)}{b} = \frac{\cos (x + 2\theta)}{c} = \frac{\cos (x + 3\theta)}{d}$,then $\left( \frac{a + c}{b + d} \right)$ is equal to:

If $O$ is the circumcentre of the $\Delta ABC$ and $R_1, R_2$ and $R_3$ are the radii of the circumcircles of triangles $OBC, OCA$ and $OAB$ respectively,then the value of $\frac{a}{R_1} + \frac{b}{R_2} + \frac{c}{R_3}$ is equal to:

Evaluate: $\sin \frac{\pi}{12} \sin \frac{2 \pi}{12} \sin \frac{3 \pi}{12} \sin \frac{4 \pi}{12} \sin \frac{5 \pi}{12} \sin \frac{6 \pi}{12}$