Question

If $f\left(x\right)=\sin \left(\log x\right)$ and $g\left(x\right)=\cos \left(\sin x\right)$ are two functions such that $\left(f\right(x){)}^2+(g\left(x\right){)}^2=1,$ then the number of maximum possible real values of $x$ is

Answer 1

$\left(f\right(x){)}^2+(g\left(x\right){)}^2=1\cdots \left(1\right)$
$\Rightarrow {\sin }^2\left(\log x\right)+{\cos }^2\left(\sin x\right)=1$
We know that ${\sin }^{2}x+{\cos }^{2}x=1$ for all real values of $x$
So, equation $\left(1\right)$ is possible only if $\log x=\sin x$


Clearly, there is only one possible solution.