If $\sin x + \text{cosec } x = 2$,then $\sin^n x + \text{cosec}^n x$ is equal to

$\left(4 \cos ^2 \frac{\pi}{20}-1\right)\left(4 \cos ^2 \frac{3 \pi}{20}-1\right)\left(4 \cos ^2 \frac{5 \pi}{20}+1\right)\left(4 \cos ^2 \frac{7 \pi}{20}-1\right)\left(4 \cos ^2 \frac{9 \pi}{20}-1\right)=$

If $\operatorname{sech}^{-1} x = \log 2$ and $\operatorname{cosech}^{-1} y = -\log 3$,then $(x + y) = $

If $\sec (\theta+\alpha), \sec \theta$ and $\sec (\theta-\alpha)$ are in arithmetic progression,then $\sin ^2 \theta=$

If $a \cos^3 \alpha + 3a \cos \alpha \sin^2 \alpha = m$ and $a \sin^3 \alpha + 3a \cos^2 \alpha \sin \alpha = n$,then $(m + n)^{2/3} + (m - n)^{2/3}$ is equal to:

If $\alpha_1, \alpha_2, \cdots, \alpha_n$ are in $A$.$P$. with common difference $\theta$, then the sum of the series $\sec \alpha_1 \sec \alpha_2 + \sec \alpha_2 \sec \alpha_3 + \cdots + \sec \alpha_{n-1} \sec \alpha_n = k(\tan \alpha_n - \tan \alpha_1)$, where $k=$