Question

If $y=e^{\{{\sin }^{2}x+{\sin }^{4}x+{\sin }^{6}x+\cdots \}}$, then $\frac{dy}{dx}=$

• A. $e^{{\tan }^{2}x}$
• B. $e^{2\tan x\cdot {\sec }^{2}x}$
• C. $e^{{\tan }^{2}x}\cdot 2\tan x{\sec }^{2}x$
• D. $e^{{\sec }^{2}x}$

Answer 1

The correct option is C $e^{{\tan }^{2}x}\cdot 2\tan x{\sec }^{2}x$

Given $y=e^{({\sin }^{2}x+{\sin }^{4}x+{\sin }^{6}x+........)}$

$\Rightarrow y=e^{\left({\sin }^{2}x\right(1+\ln y\left)\right)}$

Apply $\ln$ on both sides, we get

$\ln y=si{n}^{2}x+si{n}^{2}x\ln y$

$\Rightarrow \ln y=\frac{si{n}^{2}x}{1-si{n}^{2}x}$

$\Rightarrow y=e^{\frac{si{n}^{2}x}{1-si{n}^{2}x}}=e^{ta{n}^{2}x}$

Now differentiate wrt $x$ , we get $\frac{dy}{dx}=e^{ta{n}^{2}x}.2tanxse{c}^{2}x$

$\therefore$ option $C$ is correct