$If \sin 2\theta = k then the value of \frac{{\tan }^3\theta }{1+{\tan }^2\theta }+\frac{co{t}^3\theta }{1+co{t}^2\theta } =$

$$\begin{gathered} \sin 2\mathrm{\theta }=\mathrm{k} \\ \begin{array}{rcl} & & \frac{{\tan }^3\mathrm{\theta }}{1+{\tan }^2\mathrm{\theta }}\end{array}\begin{array}{rcl} & & +\end{array}\begin{array}{rcl} & & \frac{{\cot }^3\mathrm{\theta }}{1+{\cot }^2\mathrm{\theta }}\end{array} \\ =\frac{{\tan }^3\mathrm{\theta }}{{\sec }^2\mathrm{\theta }}+\frac{{\cot }^3\mathrm{\theta }}{{\mathrm{cosec}}^2\mathrm{\theta }} \\ =\frac{{\displaystyle \frac{{\sin }^3\mathrm{\theta }}{{\cos }^3\mathrm{\theta }}}}{{\displaystyle \frac{1}{{\cos }^2\mathrm{\theta }}}}+\frac{{\displaystyle \frac{{\cos }^3\mathrm{\theta }}{{\sin }^3\mathrm{\theta }}}}{{\displaystyle \frac{1}{{\sin }^2\mathrm{\theta }}}} \end{gathered}$$

$\begin{array}{rcl} & =& \frac{{\sin }^3\mathrm{\theta }}{\mathrm{cos\theta }}+\frac{{\cos }^3\mathrm{\theta }}{\mathrm{sin\theta }}\\ & =& \frac{{\sin }^4\mathrm{\theta }+{\cos }^4\mathrm{\theta }}{\mathrm{sin\theta } \mathrm{cos\theta }}\\ & =& \frac{\left({\sin }^2\mathrm{\theta }+{\cos }^2\mathrm{\theta }\right)-2{\sin }^2\mathrm{\theta } {\cos }^2\mathrm{\theta }}{\mathrm{sin\theta } \mathrm{cos\theta }}\\ & =& \frac{1}{\mathrm{sin\theta } \mathrm{cos\theta }}-2\mathrm{sin\theta } \mathrm{cos\theta }\\ & =& \frac{2}{\sin 2\mathrm{\theta }}-\sin 2\mathrm{\theta }\\ & =& \frac{2}{\mathrm{k}}-\mathrm{k}=\frac{2-{\mathrm{k}}^2}{\mathrm{k}}\end{array}$