$= \int\limits_0^{\frac{\pi }{8}} {\frac{{2\,\tan \theta }}{{2\tan \theta }}} \,.\,{\sec ^2}\theta \,\,d\theta \,$ $(\sin2\theta = \frac{{2\tan \theta }}{{1 + {{\tan }^2}\theta }}$ )
$= \frac{1}{a}\,.\,\frac{2}{3}\,\left[ {{{(x + a)}^{\frac{3}{2}}}\,\,{x^{\frac{3}{2}}}} \right]_0^a$ $= \left. {\tan \theta } \right|_0^{\frac{\pi }{8}}$
$= \frac{2}{{3a}}\,\left[ {{{(2a)}^{\frac{3}{2}}}\, - \,{a^{\frac{3}{2}}}\, - \,{a^{\frac{3}{2}}}} \right]\,$
$ = \,\left( {\sqrt 2 \, - \,1} \right)$
$= \frac{2}{{3a}}\,.\,2{a^{\frac{3}{2}}}\,\left[ {\sqrt 2 \, - \,1} \right]\, $
$= \,\sqrt 2 \, - \,1$
$\Rightarrow$ $\frac{4}{3}\,\sqrt a \, = \,1$
$\Rightarrow$ $a\, = \,\frac{9}{{16}}\,$
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