If $\tan A - \tan B = x$ and $\cot B - \cot A = y,$ then $\cot (A - B) = $

Let $X, Y, Z$ be respectively the areas of a regular pentagon,regular hexagon,and regular heptagon which are inscribed in a circle of radius $1$. Then,

If $\sum_{r=1}^{13} \left\{ \frac{1}{\sin \left(\frac{\pi}{4} + (r-1) \frac{\pi}{6}\right) \sin \left(\frac{\pi}{4} + \frac{r\pi}{6}\right)} \right\} = a\sqrt{3} + b$,where $a, b \in \mathbb{Z}$,then $a^2 + b^2$ is equal to:

If $\frac{\sin ^4 x}{2}+\frac{\cos ^4 x}{3}=\frac{1}{5},$ then
$(A) \tan ^2 x=\frac{2}{3}$ $(B) \frac{\sin ^8 x}{8}+\frac{\cos ^8 x}{27}=\frac{1}{125}$
$(C) \tan ^2 x=\frac{1}{3}$ $(D) \frac{\sin ^8 x}{8}+\frac{\cos ^8 x}{27}=\frac{2}{125}$

$\sin^6 \theta + \cos^6 \theta + 3 \sin^2 \theta \cos^2 \theta = $

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