Question

If $f(x) = sin\, (sin\, x)$ and $f'' (x) + tan \,x\,f'(x)+ g(x) = 0,$ then $g(x) $ is :

• A. $cos^2\, x\, cos\, (sin\, x)$
• B. $sin^2\, x\, cos\, (cos\, x)$
• C. $sin^2\, x\, sin\, (cos\, x)$
• D. $cos^2\, x\, sin\, (sin\, x)$

Answer 1

The Correct Option is D

$f \left(x\right) = sin\, \left(sin\,x\right)$
$\Rightarrow f '\left(x\right)=cos\left(sin\,x\right).cos\,x$
$\Rightarrow f ''\left(x\right)=-sin\,\left(sin\,x\right).cos^{2}\,cos\left(sin\,x\right).\left(-sin\,x\right)$
$=-cos^{2}\,x. sin\left(sin\,x\right)-sin\,x.cos\left(sin\,x\right)$
Now $f ''\left(x\right) + tan x .f '\left(x\right) + g \left(x\right) = 0$
$\Rightarrow g\left(x\right) = cos^{2}\, x . sin \left(sin\, x\right) + sin\, x . cos \left(sin\, x\right)- tan \,x . cos\, x . cos \left(sin\, x\right)$
$\Rightarrow g\left(x\right) = cos^{2}\,x . sin \left(sin\, x\right).$