$\tan 65^{\circ}, \tan 40^{\circ}+\tan 25^{\circ}$ and $\tan 25^{\circ}$ are in

If $P=\operatorname{cosec} \frac{\pi}{8}+\operatorname{cosec} \frac{2 \pi}{8}+\operatorname{cosec} \frac{3 \pi}{8}$ $+\operatorname{cosec} \frac{13 \pi}{8}+\operatorname{cosec} \frac{14 \pi}{8}+\operatorname{cosec} \frac{15 \pi}{8}$ and $\phi=8 \sin \frac{\pi}{18} \sin \frac{5 \pi}{18} \sin \frac{7 \pi}{18}$, then the value of $P+Q$ is

The minium value of $$\left[2-\cos \theta+\sin ^2 \theta\right]$$ is

If $$\sin A, \sin B$$ and $$\cos A$$ are in GP, then the roots of $$x^2+2 x \cot B+1=0$$ are always

$$\cos (x+y), \cos x, \cos (x-y)$$ are in HP, then $$\cos x \sec \frac{y}{2}$$ is

The value of $$\cos \left(\frac{3 \pi}{2}+x\right) \cos (2 \pi+x)\left\{\cot \left(\frac{3 \pi}{2}-x\right)+\cot (2 \pi+x)\right\}$$ is

If $$\theta=\frac{\pi}{2^n+1}$$, then the value of $$2^n \cos \theta \cos 2 \theta \cos 2^2 \theta \ldots \cos 2^{n-1} \theta$$ is