If $\tan A+\tan B+\tan C=\tan A·\tan B·\tan C,$ then

$\sin x, \sin 2x, \sin 3x$ are in A.P. if

The equation $2\sin \frac{x}{2}·{\cos }^{2}x+{\sin }^{2}x=2\sin \frac{x}{2}·{\sin }^{2}x+{\cos }^{2}x$ has a root for which

$5{\sin }^{2}x+\sqrt{3}\sin x\cos x+6{\cos }^{2}x=5$ if

${\sin }^{2}x+2\sin x\cos x-3{\cos }^{2}x=0$ if