If $\sin \beta$ is the GM between $\sin \alpha \text{ and }\cos \alpha ,$then $\cos 2\beta$ is equal to 

Since $\sin \beta$ is the GM between $\sin a and \cos a.$ 

$\begin{array}{l}\therefore {\sin }^2\beta =\sin \alpha \cos \alpha \\ \Rightarrow \cos 2\beta =1-2{\sin }^2\beta =1-2{\sin }^2\alpha {\cos }^2\alpha \\ \Rightarrow \cos 2\beta =(\cos \alpha -\sin \alpha {)}^2\\ \Rightarrow \cos 2\beta =2{\left\{\frac{1}{\sqrt{2}}\cos \alpha -\frac{1}{\sqrt{2}}\sin \alpha \right\}}^2\\ \Rightarrow \cos 2\beta =2{\cos }^2\left(\frac{\pi }{4}+\alpha \right)=2{\sin }^2\left(\frac{\pi }{4}-\alpha \right)\end{array}$