If $\sin (\theta + \alpha ) = a$ and $\sin (\theta + \beta ) = b,$ then $\cos 2(\alpha - \beta ) - 4ab\cos (\alpha - \beta )$ is equal to

Let $\tan \alpha, \tan \beta$ and $\tan \gamma$ (where $\alpha, \beta, \gamma \neq \frac{(2n-1)\pi}{2}, n \in N$) be the slopes of three line segments $OA, OB$ and $OC$ respectively,where $O$ is the origin. If the circumcentre of $\Delta ABC$ coincides with the origin and its orthocentre lies on the $y$-axis,then the value of $\left(\frac{\cos 3\alpha + \cos 3\beta + \cos 3\gamma}{\cos \alpha \cos \beta \cos \gamma}\right)^2$ is equal to:

The value of $\frac{\sqrt{3}\text{cosec } 20^{\circ}-\sec 20^{\circ}}{\cos 20^{\circ}\cos 40^{\circ}\cos 60^{\circ}\cos 80^{\circ}}$ is equal to

If $f(x) = \frac{\cos^2 x + \sin^4 x}{\sin^2 x + \cos^4 x}$,for $x \in R$,then $f(2002)$ is equal to

The value of $\cos 52^\circ + \cos 68^\circ + \cos 172^\circ$ is

The value of ${e^{\log_{10} \tan 1^\circ + \log_{10} \tan 2^\circ + \log_{10} \tan 3^\circ + \dots + \log_{10} \tan 89^\circ}}$ is