Question

lf $f\left(x\right)=\sin \left(\sin x\right)$ and $\cot x{f}^{{}^{\prime \prime }}\left(x\right)+f^{{}^{\prime }}\left(x\right)+g\left(x\right)\cot x=0$ then $g\left(x\right)=$

• A. $-\cos \left(\cos x\right){\sin }^{2}x$
• B. $\cos \left(\cos x\right){\sin }^{2}x$
• C. $-\sin \left(\sin x\right){\cos }^{2}x$
• D. $\sin \left(\sin x\right){\cos }^{2}x$

Answer 1

The correct option is D $\sin \left(\sin x\right){\cos }^{2}x$

$f\left(x\right)=\sin \left(\sin x\right)$

$\Rightarrow f^{\prime }\left(x\right)=\cos \left(\sin x\right)\left(\cos x\right)\Rightarrow \left(1\right)$

$\Rightarrow f^{\prime \prime }\left(x\right)=-\sin \left(\sin x\right){\cos }^{2}x-\cos \left(\sin x\right)\sin x\Rightarrow \left(2\right)$

Also given that,

$\cot x{f}^{\prime \prime }\left(x\right)+f^{\prime }\left(x\right)+g\left(x\right)\cot x=0$

$\Rightarrow g\left(x\right)=-\frac{f^{\prime }\left(x\right)}{\cot x}-f^{\prime \prime }\left(x\right)$

$=-\cos \left(\sin x\right)\sin x+\sin \left(\sin x\right){\cos }^{2}x+\cos \left(\sin x\right)\sin x=\sin \left(\sin x\right){\cos }^{2}x$

using (1) and (2).