If $\tan \theta + \cot \theta = 4$,then $\tan^{4} \theta + \cot^{4} \theta = $

If $\cos \theta - \sin \theta = \sqrt{5} \sin \theta$,then $\cos \theta + \sin \theta = $

If $\sec(\theta+\alpha)$,$\sec\theta$,and $\sec(\theta-\alpha)$ are in arithmetic progression,then the value of $\cos\theta \cdot \sec\frac{\alpha}{2}$ is:

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:

$\cot \frac{\pi}{16} \cdot \cot \frac{2 \pi}{16} \cdot \cot \frac{3 \pi}{16} \cdot \cot \frac{4 \pi}{16} \cdot \cot \frac{5 \pi}{16} \cdot \cot \frac{6 \pi}{16} \cdot \cot \frac{7 \pi}{16} = $

If $\cos^3 80^{\circ} + \cos^3 40^{\circ} - \cos^3 20^{\circ} = k$,then $\frac{4k}{3} =$