Question

If y = sin (sin x), prove that `(d^2y)/(dx^2) + tan x dy/dx + y cos^2 x = 0`

Answer 1

y = sin(sin x)

`dy/dx = cos(sin x).cos x`

`(d^2y)/(dx^2) = cos (sin x).(-sinx) + cosx {-sin(sinx)}.cos x = - sin x.cos(sin x) - y cos^2 x`

Now

`(d^2y)/(dx^2) + tanx . (dy)/(dx) + ycos^2 x`

`= -sin x. cos (sin x) - ycos^2x + sinx/cosx.cos(sin x).cosx + ycos^2x`

`= -sinx.cos(sinx) - ycos^2x + sin x.cos(sin x) + ycos^2x`

= 0

Hence proved