Plain Substitution Right? Wrong!

${ \quad \quad e }^{ cos\theta }cos(sin\theta )\\ =\quad Re\left[ { e }^{ cos\theta }(cos(sin\theta )+isin(sin\theta )) \right] \\ \quad \quad where\quad Re[.]\quad implies\quad real\quad part\quad of\quad a\quad complex\quad number.\\ =\quad Re\left[ { e }^{ cos\theta }{ .e }^{ isin\theta } \right] \\ =\quad Re\left[ { e }^{ cos\theta +isin\theta } \right] \\ =\quad Re\left[ { e }^{ { e }^{ i\theta } } \right] \\ \\ =>\int _{ 0 }^{ 2\pi }{ { e }^{ cos\theta } } cos(sin\theta )d\theta \\ =\quad Re\left[ \int _{ 0 }^{ 2\pi }{ { e }^{ { e }^{ i\theta } } } d\theta \right] \\ =\quad Re\left[ \int _{ 0 }^{ 2\pi }{ (1+\frac { { e }^{ i\theta } }{ 1! } } +\frac { { e }^{ 2i\theta } }{ 2! } +\frac { { e }^{ 3i\theta } }{ 3! } +.........\infty )d\theta \right] \\ =\quad Re\left[ \left[ \theta +\frac { { e }^{ i\theta } }{ i.1! } +\frac { { e }^{ 2i\theta } }{ 2i.2! } +\frac { { e }^{ 3i\theta } }{ 3i.3! } +..........\infty \right] \begin{matrix} 2\pi \\ 0 \end{matrix} \right] \\ =\quad Re\left[ \left[ \theta -\frac { i.{ e }^{ i\theta } }{ 1! } -\frac { i.{ e }^{ 2i\theta } }{ 2.2! } -\frac { i.{ e }^{ 3i\theta } }{ 3.3! } -.........\infty \right] \begin{matrix} 2\pi \\ 0 \end{matrix} \right] \\ =\quad \left[ \theta \right] \begin{matrix} 2\pi \\ 0 \end{matrix}\\ =\quad 2\pi$