The general value $\theta $ is obtained from the equation $\cos 2\theta = \sin \alpha ,$ is
$2\theta = \frac{\pi }{2} - \alpha $
$\theta = 2n\pi \pm \left( {\frac{\pi }{2} - \alpha } \right)$
$\theta = \frac{{n\pi + {{( - 1)}^n}\alpha }}{2}$
$\theta = n\pi \pm \left( {\frac{\pi }{4} - \frac{\alpha }{2}} \right)$
The smallest positive values of $x$ and $y$ which satisfy $\tan (x - y) = 1,\,$ $\sec (x + y) = \frac{2}{{\sqrt 3 }}$ are
The number of distinct solutions of the equation $\log _{\frac{1}{2}}|\sin x|=2-\log _{\frac{1}{2}}|\cos x|$ in the interval $[0,2 \pi],$ is
If $\cos \theta = - \frac{1}{{\sqrt 2 }}$ and $\tan \theta = 1$, then the general value of $\theta $ is
If ${\sin ^2}\theta = \frac{1}{4},$ then the most general value of $\theta $ is
The number of solutions to $\sin \left(\pi \sin ^2 \theta\right)+\sin \left(\pi \cos ^2 \theta\right)=2 \cos \left(\frac{\pi}{2} \cos \theta\right)$ satisfying $0 \leq \theta \leq 2 \pi$ is