If β is the harmonic mean of α and α , and θ is the arithmetic mean...

From question, $\sin \alpha, \sin \beta$ and $\cos \alpha$ are in H.P. then, $\sin \beta=\frac{2 \sin \alpha \cos \alpha}{\sin \alpha+\cos \alpha}$ and $\sin \alpha, \sin \theta and \cos \alpha$ are in A.P. then, $2 \sin \theta=\sin \alpha+\cos \alpha$ Statement 1: $\sqrt{2} \sin \left(\alpha+\frac{\pi}{4}\right) \cdot \sin \beta$$\begin{aligned} & =(\sin \alpha+\cos \alpha) \cdot \frac{2 \sin \alpha \cos \alpha}{\sin \alpha+\cos \alpha} \\ & =2 \sin \alpha \cos \alpha=\sin 2 \alpha \end{aligned}$$$Statement 2:$\cos \left(\alpha-\frac{\pi}{4}\right)=\frac{1}{\sqrt{2}}(\sin \alpha+\cos \alpha)=\frac{2 \sin \theta}{\sqrt{2}}=\sqrt{2} \sin \theta$