If $10 \sin^4 \theta + 15 \cos^4 \theta = 6$,then the value of $\frac{27 \operatorname{cosec}^6 \theta + 8 \sec^6 \theta}{16 \sec^8 \theta}$ is:

Suppose $\theta$ and $\phi \neq 0$ are such that $\sec(\theta + \phi)$,$\sec \theta$,and $\sec(\theta - \phi)$ are in $A.P.$ If $\cos \theta = k \cos(\frac{\phi}{2})$ for some $k$,then $k$ is equal to

$\sum_{k=1}^3 \cos^2 \left((2k-1) \frac{\pi}{12}\right)$ is equal to

If $\cos \left(\frac{\alpha-\beta}{2}\right)=2 \cos \left(\frac{\alpha+\beta}{2}\right)$,then $\tan \frac{\alpha}{2} \tan \frac{\beta}{2}=$

If $x=-\frac{1}{2}$,then $\sinh ^{-1} x+\operatorname{cosech}^{-1} x=$

If $x: y: z = \tan \left(\frac{\pi}{15}+\alpha\right): \tan \left(\frac{\pi}{15}+\beta\right): \tan \left(\frac{\pi}{15}+\gamma\right)$,then find the value of $\frac{z+x}{z-x} \sin ^2(\gamma-\alpha)+\frac{x+y}{x-y} \sin ^2(\alpha-\beta)+\frac{y+z}{y-z} \sin ^2(\beta-\gamma)$.