Explanation:

- Given: \(\tan \theta \cdot \tan 5\theta = 1\).

- This implies \(\tan 5\theta = \cot \theta\).

- Now, \(\tan (5\theta) = \frac{\sin 5\theta}{\cos 5\theta}\) and \(\cot \theta = \frac{\cos \theta}{\sin \theta}\).

- Thus, \(\frac{\sin 5\theta}{\cos 5\theta} = \frac{\cos \theta}{\sin \theta}\).

- This implies \(\sin 5\theta \cdot \sin \theta = \cos 5\theta \cdot \cos \theta\).

- Using the identity: \(\sin (A + B) = \sin A \cos B + \cos A \sin B\), we use \(A = 3\theta\) and \(B = 2\theta\).

- So, \(\sin 5\theta = \sin 3\theta \cos 2\theta + \cos 3\theta \sin 2\theta\).

- Using \(\cos 5\theta = \cos 3\theta \cos 2\theta - \sin 3\theta \sin 2\theta\).

- Both terms balance when \(\sin 2\theta = 0.5\).

- Option 2 is \(\frac{1}{2}\).

Your choice is Option:2- 1/2.