$\sqrt {2 + \sqrt {2 + 2\cos 4\theta } } = $

${\sin ^4}\frac{\pi }{8} + {\sin ^4}\frac{{3\pi }}{8} + {\sin ^4}\frac{{5\pi }}{8} + {\sin ^4}\frac{{7\pi }}{8} = $

If $\sin \theta  = \frac{1}{2}\left( {\sqrt {\frac{x}{y}\,}  + \,\sqrt {\frac{y}{x}} } \right)\,,\,\left( {x,y \in R\, - \{ 0\} } \right)$. Then

For any $\theta \, \in \,\left( {\frac{\pi }{4},\frac{\pi }{2}} \right)$, the expression $3\,{\left( {\sin \,\theta  - \cos \,\theta } \right)^4} + 6{\left( {\sin \,\theta  + \cos \,\theta } \right)^2} + 4\,{\sin ^6}\,\theta $ equals

If $(\sec A + \tan A)\,(\sec B + \tan B)\,(\sec C + \tan C)$ $ = \,(\sec A - \tan A)\,(\sec B - \tan B)\,(\sec C - \tan C),$ then each side is equal to

Let $\alpha ,\beta $ be such that $\pi < (\alpha - \beta ) < 3\pi $. If $\sin \alpha + \sin \beta = - \frac{{21}}{{65}}$ and $\cos \alpha + \cos \beta = - \frac{{27}}{{65}},$ then the value of $\cos \frac{{\alpha - \beta }}{2}$ is