What will be the angle of minimum deviation if the refracting angle o…

The correct answer is Option (4) → $(π-4θ)$

Refractive index of the prism:

$\mu = \cot \theta$

Refracting angle of prism:

$A = 2\theta$

Relation for minimum deviation:

$\mu = \frac{\sin \left(\frac{A + D_m}{2}\right)}{\sin \frac{A}{2}}$

Substitute values:

$\cot \theta = \frac{\sin \left(\frac{2\theta + D_m}{2}\right)}{\sin \theta}$

$\cot \theta = \frac{\sin (\theta + \frac{D_m}{2})}{\sin \theta}$

$\cos \theta = \sin \left(\theta + \frac{D_m}{2}\right)$

$\sin \left(\frac{\pi}{2} - \theta \right) = \sin \left(\theta + \frac{D_m}{2}\right)$

Thus, $\frac{\pi}{2} - \theta = \theta + \frac{D_m}{2}$

$\frac{D_m}{2} = \frac{\pi}{2} - 2\theta$

$D_m = \pi - 4\theta$