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Question
If y = etan x, then (cos2 x)y2 =
Options
(1 − sin 2x) y1
−(1 + sin 2x)y1
(1 + sin 2x)y1
none of these
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Solution
(c) (1 + sin 2x)y1
Here
\[y = e^{\tan x} \]
\[ \Rightarrow y_1 = e^{\tan x } \sec^2 x\]
\[ \Rightarrow y_2 = e^{\tan x } \sec^4 x + e^{\tan x} \times 2\sec x \sec x \tan x\]
\[ \Rightarrow y_2 = \sec^2 x\left( e^{\tan x} \sec^2 x + e^{\tan x } \times 2 \tan x \right)\]
\[ \Rightarrow \left( \cos^2 x \right) y_2 = y_1 + e^{\tan x } \times \frac{y_1}{\sec^2 x}2 \tan x\]
\[ \Rightarrow \left( \cos^2 x \right) y_2 = y_1 + y_1 \times 2 \sin x \cos x\]
\[ \Rightarrow \left( \cos^2 x \right) y_2 = y_1 \left( 1 + \sin2x \right)\]
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