ddx(etanx+sinx)=etanx⋅f(x)+g(x), then  (g(x))2⋅(f(x))=

# $\frac{d}{dx}\left({e}^{\mathrm{tan}x}+\mathrm{sin}x\right)={e}^{\mathrm{tan}x}\cdot f\left(x\right)+g\left(x\right)$, then  ${\left(g\left(x\right)\right)}^{2}\cdot \left(f\left(x\right)\right)=$

1. A

0

2. B

$=1$

3. C

$-1$

4. D

2

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### Solution:

$\frac{d}{dx}{e}^{\mathrm{tan}x}+\frac{d}{dx}\mathrm{sin}x={e}^{\mathrm{tan}x}\cdot {\mathrm{sec}}^{2}x+\mathrm{cos}x$

$\therefore f\left(x\right)={\mathrm{sec}}^{2}x$

$g\left(x\right)=\mathrm{cos}x$

$\therefore {\left(g\left(x\right)\right)}^{2}\cdot f\left(x\right)$

$={\mathrm{cos}}^{2}x\cdot {\mathrm{sec}}^{2}x$

$=1$

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