If $$\cos(90^\circ \theta) = \sin(\theta) \quad ( \quad 90^\circ < \theta < 90^\circ \ )$$then the value of  tan5θ is:

Given:
​$$\cos 9\theta = \sin \theta$$​​
Given the condition $$9\theta < 90^\circ$$​, all angles are acute.
Using the complementary angle identity: $$\sin\theta = \cos(90^\circ - \theta)$$​​
​$$\cos 9\theta = \cos(90^\circ - \theta)$$​​
Equating angles (because cosine is one–one for acute angles):
​$$9\theta = 90^\circ - \theta$$​​
Solving:
​$$9\theta + \theta = 90^\circ \\10\theta = 90^\circ \\\theta = 9^\circ \\Now compute \tan 5\theta: \\\tan 5\theta = \tan(5 \times 9^\circ) \\\tan 5\theta = \tan 45^\circ = 1$$​​