Question

If $fog\left(x\right)=\frac{1}{\log (\sin x{)}^2}$ and $g\left(x\right)=\sqrt{x}$, then find $f\left(x\right)$ when $x\in N$

Answer 1

Let's take the given relation,
$fog\left(x\right)=\frac{1}{\log (\sin x{)}^2}$
In place of $x$ put the value of $g\left(x\right)$ and equate it with the given relation
$fog\left(x\right)=\frac{1}{\log \left(\sin \right(\sqrt{x}{)}^2{)}^2}$
$f\left(g\right(x\left)\right)=\frac{1}{\log (\sin \left(g\right(x){)}^2{)}^2}$

Now, replacing $g\left(x\right)$ by $x$, we get,
$f\left(x\right)=\frac{1}{\log (\sin \left(x^2\right){)}^2}$