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Write the correct alternative in the following:

Question:

Write the correct alternative in the following: 

If $y=e^{\tan x}$, then $\left(\cos ^{2} x\right) y_{2}=$

A. $(1-\sin 2 x) y_{1}$

B. $-(1+\sin 2 x) y_{1}$

C. $(1+\sin 2 x) y_{1}$

D. none of these

Solution:

Given:

$y=e^{\tan x}$

$\frac{d y}{d x}=e^{\tan x}(\sec x)^{2}$

$\frac{d^{2} y}{d x^{2}}=e^{\tan x}(\sec x)^{2}(\sec x)^{2}+e^{\tan x} \times 2 \sec x \times \tan x \times \sec x$

$=e^{\tan x}(\sec x)^{2}\left[(\sec x)^{2}+2 \tan x\right]$

$\left(\cos ^{2} x\right) y_{2}=e^{\tan x}\left[(\sec x)^{2}+2 \tan x\right]$

$=e^{\tan x}\left[\frac{1+2 \sin x \cos x}{(\cos x)^{2}}\right]$

$=e^{\tan x}(\sec x)^{2}[1+2 \sin x \cos x]$

$=e^{\tan x}(\sec x)^{2}[1+\sin 2 x]$

$=[1+\sin 2 \mathrm{x}] \mathrm{y}_{1}$

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