If $\mathbf{y}\mathbf{=}\mathbf{exp}\left\{{\sin }^{2}x + {\sin }^{4}x + {\sin }^{6}x + .....\right\}$ then $\frac{dy}{dx}=$

$y=e^{{\sin }^{2}x\left(1+{\sin }^{2}x+{\sin }^{3}x+...\right)}$

Use the sum of infinite GP, the above expression can be written as  

$\begin{array}{rcl}y& =& e^{{\sin }^{2}x\left(\frac{1}{1-{\sin }^{2}x}\right)}\\ & =& e^{{\tan }^{2}x}\end{array}$

Differentiate both sides 

$\begin{array}{rcl}\frac{dy}{dx}& =& \frac{d}{dx}\left(e^{{\tan }^{2}x}\right)\\ & =& \boxed{{\mathbf{e}}^{\mathbf{t}\mathbf{a}{\mathbf{n}}^{\mathbf{2}}\mathbf{x}}\left(2tanxse{c}^{2}x\right)}\end{array}$