Question

If $\sin y=x\sin (a+y),$ then $\frac{dy}{dx}=$

• A. $\frac{\sin a}{{\sin }^2(a+y)}$
• B. $\frac{{\sin }^2(a+y)}{\sin a}$
• C. ${\sin }^2(a+y)$
• D. $\sin (a+y)$

Answer 1

The correct option is B $\frac{{\sin }^2(a+y)}{\sin a}$

Given,
$\sin y=x\sin (a+y)$
$\Rightarrow x=\frac{\sin y}{\sin (a+y)}$
$\frac{dx}{dy}=\frac{\sin (a+y)\cdot \cos y-\sin y\cdot \cos (a+y)}{{\sin }^2(a+y)}$

$\Rightarrow \frac{dx}{dy}=\frac{\sin (a+y-y)}{{\sin }^2(a+y)}=\frac{\sin \left(a\right)}{{\sin }^2(a+y)}$
$\therefore \frac{dy}{dx}=\frac{{\sin }^2(a+y)}{\sin a}$