If $\cos \left(\frac{\alpha-\beta}{2}\right)=2 \cos \left(\frac{\alpha+\beta}{2}\right)$, then $\tan \frac{\alpha}{2} \tan \frac{\beta}{2}=$

For some $a, b, c \in \mathbf{R}$, if $\sin 5 \theta=a \cos ^4 \theta \sin \theta+b \cos ^2 \theta \sin ^3 \theta+c \sin ^5 \theta$, then $a b c=$

The period of $\frac{\sin x}{\cos 3 x}+\frac{\sin 3 x}{\cos 9 x}+\frac{\sin 9 x}{\cos 27 x}+\frac{\sin 27 x}{\cos 81 x}$ is

If $\alpha=\frac{\sin ^3 x}{\cos ^2 x}, \beta=\frac{\cos ^3 x}{\sin ^2 x}$ and $\sin x+\cos x=k$, then $\alpha \sin x+\beta \cos x+3=$