If $$\theta = 9^\circ$$, then what is the value of 

$$\cot \theta \cot 2\theta \cot 3\theta \cot 4\theta \cot 5\theta \cot 6\theta \cot 7\theta \cot 8\theta \cot 9\theta$$ ?

Give, $$\theta = 9^\circ$$

$$\cot \theta \cot 2\theta \cot 3\theta \cot 4\theta \cot 5\theta \cot 6\theta \cot 7\theta \cot 8\theta \cot 9\theta$$

$$\cot 9^\circ \cot 18^\circ \cot 27^\circ \cot 36^\circ \cot 45^\circ \cot54^\circ \cot63^\circ \cot 72^\circ \cot 81^\circ$$

$$\cot 9^\circ \cot 81^\circ\cot 18^\circ\cot 72^\circ\cot 27^\circ\cot63^\circ\cot 36^\circ \cot54^\circ\cot 45^\circ$$

$$\cot(90^\circ - 81^\circ)\cot81^\circ\cot(90^\circ - 72^\circ)\cot72^\circ\cot(90^\circ - 63^\circ)\cot63^\circ\cot(90^\circ - 54^\circ)\cot54^\circ\cot45^\circ$$

$$\tan81^\circ\cot81^\circ\tan72^\circ\cot72^\circ\tan63^\circ\cot63^\circ\tan54^\circ\cot54^\circ\cot45^\circ=1$$.

As $$\tan\theta\times\cot\theta=1 and \cot45^\circ=1$$

So we get our answer as 1