If $y=\sin(\sin x)$, prove that $\dfrac{d^{2}y}{dx^{2}}+\tan x\,\dfrac{dy}{dx}+y\cos^{2}x=0$.

CBSE Class 12 · 2018 · DifferentiationLong Answer

y = sin (sin x)

, prove that

\frac{d ^{2} y}{d x ^{2}} + tan x \frac{d y}{d x} + y cos^{2} x = 0

Answer: Proved

Differentiate once (chain rule):

\frac{d y}{d x} = cos (sin x) \cdot cos x

Differentiate again (product rule):

\frac{d ^{2} y}{d x ^{2}} = - sin (sin x) cos^{2} x - cos (sin x) sin x

Now substitute into the left side, using

tan x \frac{d y}{d x} = tan x cos (sin x) cos x = sin x cos (sin x)

and

y cos^{2} x = sin (sin x) cos^{2} x

\frac{d ^{2} y}{d x ^{2}} + tan x \frac{d y}{d x} + y cos^{2} x = - sin (sin x) cos^{2} x - sin x cos (sin x) + sin x cos (sin x) + sin (sin x) cos^{2} x = 0

\frac{d ^{2} y}{d x ^{2}} + tan x \frac{d y}{d x} + y cos^{2} x = 0

CBSE Class 12 Board · 2018 · Subjective