Question

If $y=sin\left(sinx\right)$, prove that $\frac{d^{2}y}{d{x}^2}+tanx\frac{dy}{dx}+yco{s}^{2}x=0.$

Answer 1

We have $y=sin\left(sinx\right)$ $\Rightarrow \frac{dy}{dx}=cos\left(sinx\right)cosx.....\left(i\right)$

And, $\frac{d^{2}y}{d{x}^2}=-cos\left(sinx\right)sinx-sin\left(sinx\right)co{s}^{2}x$

$\Rightarrow \frac{d^{2}y}{d{x}^2}=-cos\left(sinx\right)\times \frac{sinx}{cosx}\times cosx-yco{s}^{2}x$

$\Rightarrow \frac{d^{2}y}{d{x}^2}=-tanx\times \frac{dy}{dx}-yco{s}^{2}x$ (By (i) )

Hence, $\frac{d^{2}y}{d{x}^2}+tanx\frac{dy}{dx}+yco{s}^{2}x=0$