If $$\theta$$ is a positive acute angle and $$\tan2\theta\ \tan3\theta\ =1$$, then the value of $$\theta$$ is:

Given, $$\tan2\theta\ \tan3\theta\ =1$$

$$\Rightarrow$$  $$\frac{\sin2\theta\ }{\cos2\theta\ }\frac{\sin3\theta\ }{\cos3\theta\ }\ =1$$

$$\Rightarrow$$  $$\sin2\theta\ \sin3\theta =\cos2\theta\ \cos3\theta\ \ $$

$$\Rightarrow$$  $$\cos2\theta\ \cos3\theta-\sin2\theta\ \sin3\theta=0$$

$$\Rightarrow$$  $$\cos\left(2\theta\ +3\theta\ \right)=0$$

$$\Rightarrow$$  $$\cos5\theta =0$$

$$\Rightarrow$$  $$\cos5\theta=\cos90^{\circ\ }$$

$$\Rightarrow$$  $$5\theta\ =90^{\circ\ }$$

$$\Rightarrow$$  $$\theta\ =18^{\circ\ }$$

Hence, the correct answer is Option A