Question

If y = e^{tan x}, then (cos² x)y₂ =

(a) (1 − sin 2x) y₁
(b) −(1 + sin 2x)y₁
(c) (1 + sin 2x)y₁
(d) none of these

Answer 1

(c) (1 + sin 2x)y_{1
Here,
$$\begin{gathered} y=e^{\tan x} \\ \Rightarrow y_1=e^{\tan x}{\sec }^{2}x \\ \Rightarrow y_2=e^{\tan x}{\sec }^{4}x+e^{\tan x}\times 2\sec x\sec x\tan x \\ \Rightarrow y_2={\sec }^{2}x\left(e^{\tan x}{\sec }^{2}x+e^{\tan x}\times 2\tan x\right) \\ \Rightarrow \left({\cos }^{2}x\right)y_2=y_1+e^{\tan x}\times \frac{y_1}{{\sec }^{2}x}2\tan x \\ \Rightarrow \left({\cos }^{2}x\right)y_2=y_1+y_1\times 2\sin x\cos x \\ \Rightarrow \left({\cos }^{2}x\right)y_2=y_1\left(1+\sin 2x\right) \end{gathered}$$}