$$\begin{aligned} & \overrightarrow{\mathrm{c}}=\alpha \overrightarrow{\mathrm{a}}+\beta \overrightarrow{\mathrm{b}} \ldots .(1) \\ & \overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{c}}=\alpha \overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{a}}+\beta \overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{a}} \\ & 0=\alpha+\beta \cos 15^{\circ} \ldots .(2) \\ & (1) \Rightarrow \overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{c}}=\alpha \overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{~b}}+\beta \overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{~b}} \\ & \Rightarrow \cos 75^{\circ}=\alpha \cos 15^{\circ}+\beta \ldots .(3) \\ & (2) \&(3) \Rightarrow \cos 75^{\circ}=-\beta \cos ^2 15^{\circ}+\beta \end{aligned}$$

$$\begin{aligned} & \beta=\frac{\cos 75^{\circ}}{\sin ^2 15^{\circ}}=\frac{1}{\sin 15^{\circ}}=\frac{2 \sqrt{2}}{\sqrt{3}-1} \\ & (2) \Rightarrow \alpha=\frac{-\cos 15^{\circ}}{\sin 15^{\circ}}=\frac{-(\sqrt{3}+1)}{(\sqrt{3}-1)} \\ & \therefore \overrightarrow{\mathrm{c}}=\frac{-(\sqrt{3}+1)}{(\sqrt{3}-1)} \overrightarrow{\mathrm{a}}+\left(\frac{2 \sqrt{2}}{\sqrt{3}-1}\right) \overrightarrow{\mathrm{b}} \end{aligned}$$

Now

$$\begin{aligned} & \alpha+\sqrt{2}(\sqrt{3}-1) \beta=\frac{-(\sqrt{3}+1)}{(\sqrt{3}-1)}+\frac{\sqrt{2}(\sqrt{3}-1) \cdot 2 \sqrt{2}}{\sqrt{3}-1} \\ & =\frac{-(\sqrt{3}+1)^2}{2}+4 \\ & =\frac{-3-1-2 \sqrt{3}+8}{2} \\ & =2-\sqrt{3} \end{aligned}$$