If $\sin x+\sin ^2 x=1$,where $x \in\left(0, \frac{\pi}{2}\right)$,then the value of $(\cos ^{12} x+\tan ^{12} x)+3(\cos ^{10} x+\tan ^{10} x+\cos ^8 x+\tan ^8 x)+(\cos ^6 x+\tan ^6 x)$ 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:

Let $P(\alpha, \beta)$ and $Q(\gamma, \delta)$ be two points that lie on the curve $\tan^2(x+y) + \cos^2(x+y) + y^2 + 2y = 0$ in the $XY$-plane. If the distance between $P$ and $Q$ is $d$,then $\cos d =$

If $0 \leqslant x \leqslant \pi$ and $81^{\sin ^2 x} + 81^{\cos ^2 x} = 30$,then $x$ takes the value:

If $\tan \beta = \cos \theta \tan \alpha ,$ then ${\tan ^2}\frac{\theta }{2} = $

Let $P = \{ \theta : \sin \theta - \cos \theta = \sqrt{2} \cos \theta \}$ and $Q = \{ \theta : \sin \theta + \cos \theta = \sqrt{2} \sin \theta \}$ be two sets. Then