Question

If y = e^tan x, then (cos² x)y₂ =

Options

• (1 − sin 2x) y₁

• −(1 + sin 2x)y₁

• (1 + sin 2x)y₁

• none of these

Answer 1

(c) (1 + sin 2x)y₁ 

Here

\[y = e^{\tan x} \]

\[ \Rightarrow y_1 = e^{\tan x } \sec^2 x\]

\[ \Rightarrow y_2 = e^{\tan x } \sec^4 x + e^{\tan x} \times 2\sec x \sec x \tan x\]

\[ \Rightarrow y_2 = \sec^2 x\left( e^{\tan x} \sec^2 x + e^{\tan x } \times 2 \tan x \right)\]

\[ \Rightarrow \left( \cos^2 x \right) y_2 = y_1 + e^{\tan x } \times \frac{y_1}{\sec^2 x}2 \tan x\]

\[ \Rightarrow \left( \cos^2 x \right) y_2 = y_1 + y_1 \times 2 \sin x \cos x\]

\[ \Rightarrow \left( \cos^2 x \right) y_2 = y_1 \left( 1 + \sin2x \right)\]