The value of $\cos 20^{\circ} + 2 \sin^2 55^{\circ} - \sqrt{2} \sin 65^{\circ}$ is

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 $k=$

If $\frac{2 \sin \alpha}{1+\cos \alpha+\sin \alpha}=x$,then $\frac{1-\cos \alpha-\sin \alpha}{\cos \alpha}=$

If $\tan \theta = \frac{x \sin \phi}{1 - x \cos \phi}$ and $\tan \phi = \frac{y \sin \theta}{1 - y \cos \theta}$,then $\frac{x}{y} = $

The number of pairs $(x, y)$ satisfying the equations $\sin x + \sin y = \sin (x + y)$ and $|x| + |y| = 1$ is

If $2 \sec 2\alpha = \tan \beta + \cot \beta$,then one of the values of $\alpha + \beta$ is