The integer part of the number $\sum_{k=0}^{44} \frac{1}{\cos k^{\circ} \cos (k+1)^{\circ}}$ is

Let $a, b, c$ be three non-zero real numbers such that the equation $\sqrt{3} a \cos x + 2 b \sin x = c$,where $x \in [-\frac{\pi}{2}, \frac{\pi}{2}]$,has two distinct real roots $\alpha$ and $\beta$ with $\alpha + \beta = \frac{\pi}{3}$. Then,the value of $\frac{b}{a}$ is:

If $\sin (y+z-x), \sin (z+x-y)$ and $\sin (x+y-z)$ are in $A$.$P$.,then

If $2 \sinh^{-1}\left(\frac{a}{\sqrt{1-a^2}}\right)=\log \left(\frac{1+x}{1-x}\right)$,then $x$ is equal to

Let $\alpha, \beta$ be two real numbers such that $\pi < (\alpha-\beta) < 3 \pi$. If $\sin \alpha+\sin \beta=\frac{-21}{65}$ and $\cos \alpha+\cos \beta=\frac{-27}{65}$,then $\cos \left(\frac{\beta-\alpha}{2}\right)=$

$\cos 2(\theta + \phi) - 4\cos (\theta + \phi)\sin \theta \sin \phi + 2\sin^2 \phi = $