Question

If sin y = xsin(a + y) prove that `(dy)/(dx) = sin^2(a + y)/sin a`

Answer 1

sin y = x sin(a + y)

⇒ `x = sin y/(sin (a + y))`         ....(i)

Differentiating (i) w.r.t.x,

⇒ 1 = `(sin(a + y).(d/dx sin y) - sin y. (d/dx sin (a + y)))/sin^2 (a + y)`

⇒  ` sin(a + y).cos y - d/dx - sin y. cos (a + y). d/dx = sin^2 (a + y)`]

⇒ `d/dx [ sin ( a + y) . cos y - sin y. cos ( a + y)] = sin^2 (a + y)`

⇒ `dy/dx[ sin ( a + y - y)] = sin^2 (a + y)`

⇒ `dy/dx = (sin^2 (a + y))/(sin a)`

Hence proved.