If $\underset{x\to 0}{\lim }\frac{{\tan }^{10}x-{\sin }^{10}x}{x^{10+k}}=5 then k=$

$$\begin{gathered} \underset{\mathrm{x}\to 0}{\lim }\frac{{\tan }^{10}\mathrm{x}-{\sin }^{10}\mathrm{x}}{{\mathrm{x}}^{10+\mathrm{k}}} \\ \underset{\mathrm{x}\to 0}{\lim } \frac{{\tan }^{10}\mathrm{x} (1-{\cos }^{10} \mathrm{x})}{{\mathrm{x}}^{10} {\mathrm{x}}^{\mathrm{k}} } \\ =\underset{\mathrm{x}\to 0}{\lim } \frac{{\tan }^{10}\mathrm{x}}{{\mathrm{x}}^{10}}.\frac{(1-{\cos }^2\mathrm{x})}{{\mathrm{x}}^2} \frac{(1+{\cos }^2\mathrm{x}+{\cos }^4\mathrm{x}+{\cos }^6\mathrm{x}+{\cos }^8\mathrm{x})}{{\mathrm{x}}^{\mathrm{k}-2}} \\ \mathrm{k}-2=0 \Rightarrow \mathrm{k}=2 \end{gathered}$$