Question

If siny=x sin(a+y) prove that dy/dx=sin^2(a+y)/sina

Answer 1

Answer:

Step-by-step explanation:

Siny = x Sin(a + y)

x = Siny/Sin(a + y)

Differentiating w.r.t.  y, using the Quotient Rule, we have,

dx/dy = Sin(a + y) d/dy{Siny}  - Siny. d/dy{Sin(a + y)} / Sin² (a + y)

 = Sin(a + y) * Cosy -  Siny * Cos(a + y) *d/dy(a + y), ...., / Sin² (a+y)   [Chain rule]

        = Sin(a + y) *Cosy - Siny * Cos (a + y) / Sin²(a + y)

         = Sin (a + y - y)/Sin² (a + y)

         = Sin a /  Sin² (a + y)

dy/dx = 1 / (dx/dy) = 1 / [Sin a /  Sin² (a + y)]

  = Sin² (a + y) / Sin a

 =R.H.S

Hence proved.