Question

Simplify $\frac{{\tan }^{2}x{\sec }^{2}x}{1+{\tan }^{6}x}$ and express the answer in power of tanx.

Answer 1

$\begin{array}{cc}\frac{{\tan }^{2}x{\sec }^{2}x}{1+{\tan }^{6}x}& \\ \Rightarrow \frac{{\tan }^{2}x{\sec }^{2}x}{{\left(1\right)}^3+{\left({\tan }^{2}x\right)}^3}& x^3+y^3=\left(x+y\right)\left(x^2-xy+y^2\right)\\ \Rightarrow \frac{{\tan }^{2}x{\sec }^{2}x}{\left(1+{\tan }^{2}x\right)\left({\tan }^{4}x-{\tan }^{2}x+1\right)}& \\ \Rightarrow \frac{{\tan }^{2}x{\sec }^{2}x}{{\sec }^{2}x\left({\tan }^{4}x-{\tan }^{2}x+1\right)}& \\ \Rightarrow \frac{{\tan }^{2}x}{{\tan }^{4}x-{\tan }^{2}x+1}& \end{array}$