$$ \begin{aligned} & \int_0^{\pi / 3} \frac{\tan \theta}{\sqrt{2 \mathrm{k} \sec \theta}} \mathrm{~d} \theta \\ & =\frac{1}{\sqrt{2 \mathrm{k}}} \int_0^{\pi / 3} \frac{\sin \theta}{\sqrt{\cos \theta}} \mathrm{~d} \theta \end{aligned} $$ _et $\cos \theta=\mathrm{t}^2$ $\therefore \sin \theta d \theta=-2 t d t$ $$ \begin{aligned} & =\frac{1}{\sqrt{2 k}} \int_1^{\sqrt{\frac{1}{2}}} \frac{-2 t d t}{t} \\ & =\sqrt{\frac{2}{k}} \int_{\frac{1}{\sqrt{2}}}^1 d t \\ & =\sqrt{\frac{2}{k}}\left(1-\frac{1}{\sqrt{2}}\right) \\ & =\frac{\sqrt{2}-1}{\sqrt{k}} \\ & =1-\frac{1}{\sqrt{2}} \quad \text { (Given) } \end{aligned} $$ $$ \because k=2 $$