Step-by-step Solution:

We are given that \(\sin \beta\) is the geometric mean (GM) of \(\sin \alpha\) and \(\cos \alpha\). This means: \[ \sin \beta = \sqrt{\sin \alpha \cdot \cos \alpha} \] We need to find \(\cos 2\beta\). Using the double-angle identity: \[ \cos 2\beta = 1 - 2 \sin^2 \beta \] Substituting \(\sin \beta = \sqrt{\sin \alpha \cos \alpha}\): \[ \sin^2 \beta = \sin \alpha \cos \alpha \] So, \[ \cos 2\beta = 1 - 2 \sin \alpha \cos \alpha \] Using the identity: \[ \sin \alpha \cos \alpha = \frac{1}{2} \sin 2\alpha \] we get: \[ \cos 2\beta = 1 - 2 \times \frac{1}{2} \sin 2\alpha = 1 - \sin 2\alpha \] Now, using another identity: \[ 1 - \sin 2\alpha = 2 \sin^2 \left(\frac{\pi}{4} - \alpha \right) \] Thus, \[ \cos 2\beta = 2 \sin^2 \left(\frac{\pi}{4} - \alpha \right) \]