If $\sqrt 3 \cos \,\theta + \sin \theta = \sqrt 2 ,$ then the most general value of $\theta $ is
$n\pi + {( - 1)^n}\frac{\pi }{4}$
${( - 1)^n}\frac{\pi }{4} - \frac{\pi }{3}$
$n\pi + \frac{\pi }{4} - \frac{\pi }{3}$
$n\pi + {( - 1)^n}\frac{\pi }{4} - \frac{\pi }{3}$
$cos (\alpha \,-\,\beta ) = 1$ and $cos (\alpha +\beta ) = 1/e$ , where $\alpha , \beta \in [-\pi , \pi ]$ . Number of pairs of $(\alpha ,\beta )$ which satisfy both the equations is
Let $S\, = \,\left\{ {\theta \, \in \,[ - \,2\,\pi ,\,\,2\,\pi ]\, :\,2\,{{\cos }^2}\,\theta \, + \,3\,\sin \,\theta \, = \,0} \right\}$. Then the sum of the elements of $S$ is
If $\tan (\cot x) = \cot (\tan x),$ then $\sin 2x =$
Let $S=\left[-\pi, \frac{\pi}{2}\right)-\left\{-\frac{\pi}{2},-\frac{\pi}{4},-\frac{3 \pi}{4}, \frac{\pi}{4}\right\}$. Then the number of elements in the set $=\{\theta \in S : \tan \theta(1+\sqrt{5} \tan (2 \theta))=\sqrt{5}-\tan (2 \theta)\}$ is $...$
The number of solutions of the equation $\sin x=$ $\cos ^{2} x$ in the interval $(0,10)$ is