$sin^{2n}x + cos^{2n}x$ lies between

If $\cos p\theta = \cos q\theta ,p \ne q$, then

If $0 \le x < 2\pi $ , then the number of real values of $x,$ which satisfy the equation  $\cos x + \cos 2x + \cos 3x + \cos 4x = 0$ is  . .  .

If $\alpha ,\,\beta ,\,\gamma $ and $\delta $ are the solutions of the equation $\tan \left( {\theta  + \frac{\pi }{4}} \right) = 3\,\tan \,3\theta $ , no two of which have equal tangents, then the value of $tan\, \alpha  + tan\, \beta + tan\, \gamma + tan\, \delta $ is

Common roots of the equations $2{\sin ^2}x + {\sin ^2}2x = 2$ and $\sin 2x + \cos 2x = \tan x,$ are

Let $A=\left\{\theta \in R:\left(\frac{1}{3} \sin \theta+\frac{2}{3} \cos \theta\right)^2=\frac{1}{3} \sin ^2 \theta+\frac{2}{3} \cos ^2 \theta\right\}$.Then