Question

If $\sqrt{\tan y}=e^{\cos 2x}\sin x$, then $\frac{dy}{dx}$ is equal to

• A. $\sin 2y(\cot x+2\sin 2x)$
• B. $\sin 2y(\cot x-2\sin 2x)$
• C. $\sin 2x(\cot x-2\sin 2x)$
• D. $\sin 2x(\cot x+2\sin 2x)$

Answer 1

The correct option is B $\sin 2y(\cot x-2\sin 2x)$

Given,
$\sqrt{\tan y}=e^{\cos 2x}\sin x$
Applying logarithm both sides,
$\Rightarrow \frac{1}{2}\ln \left(\tan y\right)=\cos 2x+\ln \left(\sin x\right)$
Differentiating w.r.t. $x,$
$\Rightarrow \frac{1}{2}\cdot \frac{1}{\tan y}\cdot {\sec }^{2}y\left(\frac{dy}{dx}\right)=-2\sin 2x+\frac{1}{\sin x}\cdot \cos x$
$\Rightarrow \frac{1}{2\sin y\cos y}\cdot \frac{dy}{dx}=\cot x-2\sin 2x$
$\Rightarrow \frac{dy}{dx}=\sin 2y(\cot x-2\sin 2x)$