Question

If $y=\sin \left(\sin x\right)$, then the value of $y^{\prime \prime }+\left(\tan x\right)y^{\prime }+y{\cos }^{2}x+5$ is

Answer 1

$y=\sin \left(\sin x\right)\Rightarrow y^{\prime }=\cos \left(\sin x\right)\cdot \cos x\Rightarrow y^{\prime \prime }=-\sin \left(\sin x\right)\cdot {\cos }^{2}x-\cos \left(\sin x\right)\cdot \sin x\Rightarrow y^{\prime \prime }=-y{\cos }^{2}x-\sin x\cdot \frac{y^{\prime }}{\cos x}\Rightarrow y^{\prime \prime }+\left(\tan x\right)y^{\prime }+y{\cos }^{2}x=0\therefore y^{\prime \prime }+\left(\tan x\right)y^{\prime }+y{\cos }^{2}x+5=5$