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

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

A

sin⁡asin⁡2(a+y)\frac{\sin a}{\sin^{2}(a + y)}

B

sin⁡2(a+y)sin⁡a\frac{\sin^{2}(a + y)}{\sin a}

C

cos⁡asin⁡2(a+y)\frac{\cos a}{\sin^{2}(a + y)}

D

sin⁡2(a+y)cos⁡a\frac{\sin^{2}(a + y)}{\cos a}

Answer

sin⁡2(a+y)sin⁡a\frac{\sin^{2}(a + y)}{\sin a}

Explanation

Solution

x = sin y / sin (a + y)

dxdy=sin⁡(a+y)cos⁡y−sin⁡ycos⁡(a+y)sin⁡2(a+y)\frac{dx}{dy} = \frac{\sin(a + y)\cos y - \sin y\cos(a + y)}{\sin^{2}(a + y)}

= sin⁡((a+y)−y)sin⁡2(a+y)\frac{\sin((a + y) - y)}{\sin^{2}(a + y)} = dydx=sin⁡2(a+y)sin⁡a\frac{dy}{dx} = \frac{\sin^{2}(a + y)}{\sin a}